There simply is a difference in meaning between the two words, and that difference in meaning explains also why they are more often used in specific situations.
Everything means "all things", whereas anything means "any one thing".
He will eat anything.
Whatever kind of food you give him, he will eat it. There are no things that he will not eat.
He will eat everything.
There will be no food left for others: he will eat all the food.
He will not eat anything.
There is not a single thing that he will eat -> he eats nothing.
He will not eat everything.
He will eat, but he will leave some food uneaten. (It may be that it is too much, it may be that he doesn't want to eat specific things.)
Will he eat everything?
You are asking if you will have any food left after he has finished eating.
Will he eat anything?
You are asking if he will be hungry.