David Moyes' men conceded three goals in the space of four first-half minutes as the Latics shattered their hopes of contesting a semi-final at Wembley.
And Nevilla lambasted his team's performance, insisting they need to take the criticism coming their way and use the defeat as a catalyst to achieve a top-four finish in the Premier League.
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"It wasn't good enough as a captain and all the lads feel the same.
"We can't hide away from the fact that we have to take our medicine now, take the criticism that comes our way.
"We really felt this was going to be our year in the FA Cup but we were beaten on the day by a team who were better than us.
"We have got two months left in the season and we have to bounce back and get a reaction.
"That's what the manager wants and I'm confident in the group of players that I share a dressing room with that we will do that."
"The Champions League place is in the distance. A European place, we would take that now.
"It's what we have been going for all season, making sure that next season there is European football being played at Everton. That's our aim and remains our aim.
"If we can finish in the top four then so be it, but at the moment our short-term aim is to win next Saturday against Manchester City and to attack the last eight or nine games of the season."
The defeat raised fresh speculation that Moyes could seek a way out of Goodison Park this summer, but Neville refused to comment on his manager's future.
"That's something that only the gaffer can answer," he insisted. "He has brought all of the players in the squad to the club and we all want him to stay but that's up to the manager.
"We aren't using that as an excuse for Saturday's performance. If anyone does, that's a bit of a cop out."