I'm really lucky. My daughter isn't fussy about food. She'll eat anything.
I don't cause problems when we go out for dinner. I eat anything.
As you can see, either the present tense or the use of "will eat" meaning "is happy to eat" or "is willing to eat" is acceptable.
I would say that "anything" is usually more appropriate in most cases. However, "everything" could be used to be particularly emphatic especially when it's almost true.
I have a 2-month old puppy at home. It's a nightmare. When I go out, she eats everything in the house.
In that example, it probably does mean that the puppy eats/chews a lot of non-food items - the furniture, clothing, books etc.