To be fair to Wilshere, he's only 22. Surely, it's too early to draw any conclusions about psychological issues, especially not just after a game in which his team was playing without their three biggest goal threats, their best player this season, and were facing one of the strongest club sides in recent times. They also had to contend with being down to 10 men for most of the game (may as well have been 9 tbh).
He's got a lot of responsibility at Arsenal for such a young player and is under a lot of pressure whenever he plays for England. It wouldn't surprise me if he wasn't able to cope with the pressure, but that would be more indicative of England's shortcomings and Arsenal's injury problems than anything else. He shouldn't be under so much pressure at such an early stage in his development tbh. Rooney was in a similar position at a younger age and, inevitably, couldn't live up to the ridiculous expectations people had for him, regardless of the fact that he developed into a top class player.
As a football player you'll always be under pressure. I don't think it was any easier for Isco, Götze, Draxler or Kroos.