***** NOT A TEACHER *****
Good morning, Chanhwee:
You have asked a great question.
1. I did some googling, and I think that if you wish to make your teacher happy, you should use "healthily."
a. The usual reason is quite clear: How to eat? In a healthy manner. "Healthily," as you know, is an adverb that means "in a healthy manner." It modifies the verb "eat."
2. Here in the United States, many people simply say, "Eat healthy." Why?
a. Well, some people think that it is a short way to say "Eat healthy food."
i. Remember Mr. Steve Jobs's famous "Think different"?
3. I found a very good excuse for using "healthy" instead of "healthily." It comes from an expert named Jan Freeman:
"Eat healthy" isn't missing an adverb; it just happens to have borrowed "healthy," the adjective form, to serve
in place of "healthily" or "healthfully." That doesn't make "healthy" an adjective though; it's the job, not the
Many years ago, a famous leader said something like this: It doesn't matter whether the cat is black or white; the only thing that
matters is whether the cat catches the mice.
So Ms. Freeman is telling us that if "healthy" is being used as an adverb, then it is an adverb.
In summary (My opinions only):
a. If you want to use "book English," follow the rule and use "healthily."
b. If you want to speak like many Americans, use "healthy."
c. If you want to make almost everyone happy, use "healthfully."
Source of Ms. Freeman's opinion: Mortal Syntax by Ms. June Casagrande (Google books).