From a technical point of view, the correct phrase is "eat healthily" as "healthily" is an adverb that modifies the meaning of the verb "eat". Healthy is an adjective that would be used to describe something about a noun. Please note that the phrase "eat healthy" is what is known as an acceptable error in English, so if you were to say "eat healthy" instead of "eat healthily" in conversation to an English speaker they are likely to not even notice.