Both are correct but have slightly different meanings. Anything and everything are synonyms but can be used to express different thoughts depending on the context. Saying I eat anything basically means I am not fussy. If you say I eat everything you could be saying I am not fussy or you could be saying you eat literally everything as in you eat a lot.

I won't necessarily eat anything, though most things yes hehe. And depending on how much food is being served, I might eat everything.

I eat everything is correct ."I don't eat anything" is the correct form of negative.

The other answers are good. I will just add: I eat anything = I don't care, I don't have any preferences. I eat everything = I love all food. If it is food, I love it! Good luck!

I would add that 'anything' is generally a kind of 'everything' which we don't know what it is, OR, 'something' that is not there. Basically, when there is uncertainty, we use anything. Examples: 1a) I've already eaten everything in my fridge. (specific things that we know exist). 1b) I'm so hungry right now, I could eat anything! (everything that is possible, but we don't specifically know what it is). 2a) I'm going to order something to eat (a positive statement, so we use 'something'). 2b) I'm not going to order anything to eat. (a negative statement, something but it is not there, so we use 'anything').