Yes: one is wrong and one is right!
"I eat everything" is correct English. "Everything" is a pronoun, so it can be used as the object of a sentence : I see everything, I understand everything and so on.
"I eat all" is not correct English. In most cases in modern English, 'all' is used as a determiner. Determiners cannot stand alone as subjects or objects of sentences: they have to be used together with nouns and pronouns. For example, you can say "I eat all the food" or "I eat all kinds of things". You cannot say "I eat all".