Arsenal midfielder Wilshere played only 141 minutes in the Premier League this season after breaking a leg, while Liverpool’s Henderson was out for five weeks with a knee injury. before returning as a substitute in their final league game at West Brom.
Match of the Day pundit Shearer, who scored 30 goals in 63 games for England between 1992 and 2000, said: “Once again England have gone for reputation rather than form. I hope the manager doesn’t regret it.”
The verdict of BBC Sport’s England Euro 2016 team selector – after it passed 1m users – matches Shearer’s.
Although BBC Sport users are asked to pick an England XI, rather than a 23-man squad, one third of people (358,000) chose Drinkwater in their team.
He came fifth for midfield selections behind Spurs’ duo Dele Alli (900,000) and Eric Dier (673,000), Manchester United and England captain Wayne Rooney (580,000) and Manchester City winger Raheem Sterling (378,000).
A Premier League champion with Leicester City this season, Drinkwater was a more popular midfield choice than Everton’s Ross Barkley (322,000), Liverpool trio James Milner (286,000), Adam Lallana (282,000) and Jordan Henderson (230,000), and Arsenal’s Jack Wilshere (253,000).
On Hodgson’s other two omissions, BBC Sport users largely agree, with City’s Fabian Delph (34,000) and Newcastle winger Andros Townsend (175,000) the bottom two midfield picks.
But few users chose teenage striker Marcus Rashford (116,000) to start for England in France, even though the Manchester United player scored on his international debut against Australia on Friday.
Shearer said that the 18-year-old deserves to be in the squad because of his performances in the last three months but said he would be “very surprised” if he started.
“If needs be, he can come and make an impact if England are chasing the game,” said Shearer. “Once he gets on, he may force the manager’s hand.”
According to data from the 1m uses so far, a BBC Sport team selector XI for England’s first game against Russia in Marseille on 11 June, would be:
Agree with Shearer? Is this England team is right? Have a go and share your #MyEnglandXI
Who do you think should start at Euro 2016? Step into Roy Hodgson’s shoes and pick your XI – and then share it with your friends using our brand new team selector.
Subscribe to the BBC Sport newsletter to get our pick of news, features and video sent to your inbox.
Bayern Munich’s David Alaba and Leicester City’s Christian Fuchs are the star names in Marcel Koller’s squad after winning league titles in Germany and England respectively.
Goalkeepers: Robert Almer (Austria Vienna), Heinz Lindner (Eintracht Frankfurt), Ramazan Oezcan (Ingolstadt).
Defenders: Aleksandar Dragovic (Dynamo Kiev), Christian Fuchs (Leicester City), Gyorgy Garics (Darmstadt), Martin Hinteregger (Borussia Moenchengladbach), Florian Klein (Stuttgart), Sebastian Prodl (Watford), Markus Suttner (Ingolstadt), Kevin Wimmer (Tottenham Hotspur).
Midfielders: David Alaba (Bayern Munich), Marko Arnautovic (Stoke City), Julian Baumgartlinger (Mainz), Martin Harnik (Stuttgart), Stefan Ilsanker (RB Leipzig), Jakob Jantscher (Lucerne), Zlatko Junuzovic (Werder Bremen), Marcel Sabitzer (RB Leipzig), Alessandro Schoepf (Schalke).
Forwards: Lukas Hinterseer (Ingolstadt), Rubin Okotie (1860 Munich), Marc Janko (Basel).
Veteran defender Vedran Corluka makes the cut despite suffering an ankle injury in training on Tuesday, but Croatia’s hopes will largely rest on the form of influential midfielders Luka Modric and Ivan Rakitic.
Defenders: Vedran Corluka (Lokomotiv Moscow), Darijo Srna (Shakhtar Donetsk), Domagoj Vida (Dynamo Kiev), Sime Vrsaljko (Sassuolo), Gordon Schildenfeld (Dinamo Zagreb), Ivan Strinic (Napoli), Tin Jedvaj (Bayer Leverkusen).
Midfielders: Luka Modric, Mateo Kovacic (both Real Madrid), Ivan Rakitic (Barcelona), Marcelo Brozovic (Inter MIlan), Ivan Perisic (Inter Milan), Milan Badelj (Fiorentina), Marko Rog (Dinamo Zagreb), Ante Coric (Dinamo Zagreb).
Forwards: Mario Mandzukic (Juventus), Nikola Kalinic (Fiorentina), Marko Pjaca (Dinamo Zagreb), Duje Cop (Dinamo Zagreb), Andrej Kramaric (Hoffenheim).
Tomas Rosicky is included despite playing just 19 minutes for Arsenal this season because of injuries.
Goalkeepers: Petr Cech (Arsenal), Tomas Vaclik (Basel), Tomas Koubek (Sparta Prague).
Defenders: Theodor Gebre Selassie (Werder Bremen), Pavel Kaderabek (Hoffenheim), David Limbersky (Viktoria Pilsen), Marek Suchy (Basel), Michal Kadlec (Sparta Prague), Tomas Sivok (Bursaspor), Roman Hubnik (Viktoria Pilsen), Daniel Pudil (Sheffield Wednesday).
Midfielders: Borek Dockal (Sparta Prague), Jiri Skalak (Brighton), Vladimir Darida (Hertha Berlin), Daniel Kolar (Viktoria Pilsen), Ladislav Krejci (Sparta Prague), Josef Sural (Sparta Prague), David Pavelka (Kasimpasa), Jaroslav Plasil (Bordeaux), Tomas Rosicky (Arsenal).
Forwards: Milan Skoda (Slavia Prague), Tomas Necid (Bursaspor), David Lafata (Sparta Prague).
Manchester United’s 18-year-old striker Marcus Rashford, who scored on his international debut, makes Roy Hodgson’s 23, but there is no place for Leicester City midfielder Danny Drinkwater.
Goalkeepers: Joe Hart (Manchester City), Fraser Forster (Southampton), Tom Heaton (Burnley).
Defenders: Gary Cahill (Chelsea), Chris Smalling (Manchester United), John Stones (Everton), Kyle Walker (Tottenham Hotspur), Ryan Bertrand (Southampton), Danny Rose (Tottenham Hotspur), Nathaniel Clyne (Liverpool).
Midfielders: Dele Alli (Tottenham Hotspur), Ross Barkley (Everton), Eric Dier (Tottenham Hotspur), Jordan Henderson (Liverpool), Adam Lallana (Liverpool), James Milner (Liverpool), Raheem Sterling (Manchester City), Jack Wilshere (Arsenal).
Strikers: Wayne Rooney (Manchester United), Harry Kane (Tottenham Hotspur), Jamie Vardy (Leicester City), Daniel Sturridge (Liverpool), Marcus Rashford (Manchester United).
Former West Brom and Fulham midfielder Zoltan Gera, 37, will feature in Hungary’s first appearance at a European Championship since 1972.
Goalkeepers: Gabor Kiraly (Haladas), Denes Dibusz (Ferencvaros), Peter Gulacsi (Leipzig).
Defenders: Attila Fiola (Puskas Academy), Barnabas Bese (MTK Budapest), Richard Guzmics (Wisla Krakow), Roland Juhasz (Videoton), Adam Lang (Videoton), Tamas Kadar (Lech Poznan), Mihaly Korhut (Debrecen).
Midfielders: Adam Pinter (Ferencvaros), Gergo Lovrencsics (Lech Poznan), Akos Elek (Diosgyor), Zoltan Gera (Ferencvaros), Adam Nagy (Ferencvaros), Laszlo Kleinheisler (Werden Bremen), Zoltan Stieber (Hamburg).
Midfielders: William Carvalho (Sporting), Danilo Pereira (Porto), Joao Moutinho (Monaco), Renato Sanches (Bayern Munich), Adrien Silva (Sporting), Andre Gomes (Valencia), Joao Mario (Sporting).
Forwards: Rafa Silva (Braga), Ricardo Quaresma (Besiktas), Nani (Fenerbahce), Cristiano Ronaldo (Real Madrid), Eder (Lille).
Republic of Ireland
Veteran striker Robbie Keane suffered a calf injury last week but has recovered to captain Martin O’Neill’s side in France.
Goalkeepers: Darren Randolph (West Ham), Shay Given (Stoke), Keiren Westwood (Sheffield Wednesday).
Defenders: Cyrus Christie (Derby), Seamus Coleman (Everton), Ciaran Clark (Aston Villa), Richard Keogh (Derby), Shane Duffy (Blackburn), John O’Shea (Sunderland), Stephen Ward (Burnley).
Midfielders: James McClean (West Brom), Glenn Whelan (Stoke), James McCarthy (Everton), Jeff Hendrick (Derby), Stephen Quinn (Reading), Wes Hoolahan (Norwich), David Meyler (Hull City), Robbie Brady (Norwich), Aiden McGeady (Everton).
Forwards: Jon Walters (Stoke), Shane Long (Southampton), Robbie Keane (LA Galaxy), Daryl Murphy (Ipswich).
Schalke’s defensive midfielder Roman Neustadter is the only player to be included in the squad who plays his football outside Russia.
Goalkeepers: Igor Akinfeev (CSKA Moscow), Yuri Lodygin (Zenit St Petersburg), Guilherme (Lokomotiv Moscow).
Defenders: Aleksei Berezutski (CSKA Moscow), Vasili Berezutski (CSKA Moscow), Sergei Ignashevich (CSKA Moscow), Dmitri Kombarov (Spartak Moscow), Roman Neustadter (Schalke), Georgi Schennikov (CSKA Moscow), Roman Shishkin (Lokomotiv Moscow), Igor Smolnikov (Zenit St Petersburg).
Midfielders: Igor Denisov (Dynamo Moscow), Dmitri Torbinski (Krasnodar), Aleksandr Golovin (CSKA Moscow), Denis Glushakov (Spartak Moscow), Oleg Ivanov (Terek Grozny), Pavel Mamaev (Krasnodar), Aleksandr Samedov (Lokomotiv Moscow), Oleg Shatov (Zenit St Petersburg), Roman Shirokov (CSKA Moscow).
Forwards: Artem Dzyuba (Zenit St Petersburg), Aleksandr Kokorin (Zenit St Petersburg), Fedor Smolov (Krasnodar).
Napoli playmaker Marek Hamsik is expected to be the danger man for Slovakia, who are in Group B alongside England and Wales.
Goalkeepers: Matus Kozacik (Viktoria Plzen), Jan Mucha (Slovan Bratislava), Jan Novota (Rapid Vienna).
Defenders: Peter Pekarik (Hertha Berlin), Milan Skriniar (Sampdoria), Martin Skrtel (Liverpool), Norbert Gyomber (Roma), Jan Durica (Lokomotiv Moscow), Kornel Salata (Slovan Bratislava), Tomas Hubocan (Dinamo Moscow), Dusan Svento (Cologne).
Midfielders: Marek Hamsik (Napoli), Juraj Kucka (AC Milan), Miroslav Stoch (Bursaspor), Vladimir Weiss (Al-Gharafa), Robert Mak (PAOK), Patrik Hrosovsky (Viktoria Plzen), Jan Gregus (Jablonec), Viktor Pecovsky (Zilina), Stanislav Sestak (Ferencvaros), Ondrej Duda (Legia Warsaw).
Forwards: Michal Duris (Viktoria Plzen), Adam Nemec (Willem II).
Striker Lucas Vazquez, who scored a penalty in Real Madrid’s Champions League triumph, has never played for Spain but makes the squad at the expense of Isco Alarcon.
Goalkeepers: Iker Casillas (Porto), David de Gea (Manchester United), Sergio Rico (Sevilla).
Defenders: Marc Bartra (Barcelona), Jordi Alba (Barcelona), Gerard Pique (Barcelona), Cesar Azpilicueta (Chelsea), Hector Bellerin (Arsenal), Sergio Ramos (Real Madrid), Juanfran (Atletico Madrid).
Midfielders: Sergio Busquets (Barcelona), Mikel San Jose (Athletic Bilbao), Cesc Fabregas (Chelsea), Thiago Alcantara (Bayern Munich), Bruno Soriano (Villarreal), Andres Iniesta (Barcelona), Koke (Atletico Madrid).
Forwards: Pedro (Chelsea), David Silva (Manchester City), Alvaro Morata (Juventus), Aritz Aduriz (Athletic Bilbao), Nolito (Celta Vigo), Lucas Vazquez (Real Madrid).
Sweden have included six players from the team that won the European Under-21 Championship last year, as well as 34-year-old Zlatan Ibrahimovic.
Goalkeepers: Andreas Isaksson (Kasimpasa), Robin Olsen (Copenhagen), Patrik Carlgren (AIK).
Defenders: Ludwig Augustinsson (Copenhagen), Erik Johansson (Copenhagen), Pontus Jansson (Torino), Victor Lindelof (Benfica) Andreas Granqvist (Krasnodar), Mikael Lustig (Celtic), Martin Olsson (Norwich).
Midfielders: Jimmy Durmaz (Olympiakos), Albin Ekdal (Hamburg), Oscar Hiljemark (Palermo), Sebastian Larsson (Sunderland), Pontus Wernbloom (CSKA Moscow), Erkan Zengin (Trabzonspor), Oscar Lewicki (Malmo), Emil Forsberg (Leipzig), Kim Kallstrom (Grasshoppers).
Forwards: Marcus Berg (Panathinaikos), John Guidetti (Celta Vigo), Zlatan Ibrahimovic (Paris St-Germain), Emir Kujovic (Norrkoping).
New Arsenal signing Granit Xhaka is included in the Switzerland squad but former Gunners defender Philippe Senderos, 31, misses out.
Goalkeepers: Roman Burki (Borussia Dortmund), Marwin Hitz (Augsburg), Yann Sommer (Monchengladbach).
Defenders: Johan Djourou (Hamburg), Nico Elvedi (Monchengladbach), Michael Lang (Basel), Stephan Lichtsteiner (Juventus), Francois Moubandje (Toulouse), Ricardo Rodríguez (Wolfsburg), Fabian Schar (Hoffenheim), Steve von Bergen (Young Boys).
Defenders: Gokhan Gonul (Fenerbahce), Ahmet Calik (Genclerbirligi), Sener Ozbayrakli (Fenerbahce), Hakan Balta (Galatasaray), Mehmet Topal (Fenerbahce), Semih Kaya (Galatasaray), Ismail Koybasi (Besiktas), Caner Erkin (Fenerbahce).
Midfielders: Emre Mor (FC Nordsjaelland), Volkan Sen (Fenerbahce), Hakan Calhanoglu (Leverkusen), Nuri Sahin (Borussia Dortmund), Oguzhan Ozyakup (Besiktas), Ozan Tufan (Fenerbahce), Selcuk Inan (Galatasaray), Arda Turan (Barcelona), Olcay Sahan (Besiktas).
Forwards: Roman Zozulya (Dnipro), Pylyp Budkivskyi (Shakhtar), Yevhen Seleznyov (Shakhtar).
Crystal Palace midfielder Joe Ledley is selected after returning to training less than a month after breaking his leg.
Goalkeepers: Wayne Hennessey (Crystal Palace), Daniel Ward (Liverpool), Owain Fon Williams (Inverness Caledonian Thistle).
Defenders: Ashley Williams (Swansea City), James Chester (West Bromwich Albion), Ben Davies (Tottenham Hotspur), James Collins (West Ham United), Chris Gunter (Reading), Neil Taylor (Swansea City), Jazz Richards (Fulham).
Midfielders: Joe Ledley (Crystal Palace), Joe Allen (Liverpool), David Vaughan (Nottingham Forest), Jonathan Williams (Crystal Palace), David Edwards (Wolverhampton Wanderers), George Williams (Fulham), Aaron Ramsey (Arsenal), Andy King (Leicester City).
Forwards: Gareth Bale (Real Madrid), David Cotterill (Birmingham City), Hal Robson-Kanu (Reading), Simon Church (MK Dons), Sam Vokes (Burnley).