Looking their' recent performance, I don't think they will relegated. I believe they will comeback, but after all Everton is just Everton, they can't going up to reach top 7 and will end near bottom position.
However I remember before this season started, many people were praise Burnley and they think Burnley could be a new contender, but they're currently stay at the lowest position lol.
Everton will definitely bounce back from their set back though it's really going to affect them big time and tell on their results they will be having on the nearest future, they are most likely to get out of relegation soonest, I'd they can continually perform brilliantly as they have in the recent past.
Their next match is against Manchester united, if they are really good, considering the recent form Manchester united has, they will be able to get a point from that match and if possible get full points if they are able to perform better than Manchester united will do, that will really had greatly to their chances of leaving the temporary relegation line almost immediately,.but then they are very likely to stay out relegation before the season ends.