Roy Hodgson's side begin their campaign against Russia on 11 June, before facing Wales and Slovakia.
"Forget who's going to cost us," Owen, who represented England at five major tournaments, told BBC Sport.
"There's almost a fear of being 'the one' - and that then turns into a selfish attitude. You can't be expressive, you can't be creative."
In an interview with BBC Sport's David Ornstein, the 36-year-old added: "There will be one person that everyone's going to gang up on - the press, the media, whatever - but you've got to be bigger than that, not bothered about that and actually just do what's best for the team.
"There's been a fear over the years because we all see someone missing a penalty or someone getting sent off and costing the team.
"The whole country then almost gangs up on that player.
"There's a fear amongst the squad - or there certainly was in my time - whereby you just didn't want to be that person everybody castigates, gets vilified, and you're the one that misses a penalty and they make a pizza advert about you."
Gareth Southgate featured in an advert for Pizza Hut after Euro 96 following England's semi-final exit to eventual tournament winners Germany.
The host nation went out after the then Aston Villa defender was the only player to miss a penalty in the shootout after the game at Wembley had finished 1-1 following extra time.
"There's been a fear over the years because we all see someone missing a penalty or someone getting sent off and costing the team.
"The whole country then almost gangs up on that player.
"There's a fear amongst the squad - or there certainly was in my time - whereby you just didn't want to be that person everybody castigates, gets vilified, and you're the one that misses a penalty and they make a pizza advert about you."
Gareth Southgate featured in an advert for Pizza Hut after Euro 96 following England's semi-final exit to eventual tournament winners Germany.
The host nation went out after the then Aston Villa defender was the only player to miss a penalty in the shootout after the game at Wembley had finished 1-1 following extra time.
