Why would "eat healthy" be incorrect? This usage of "healthy" as an adverb is already in at least one major dictionary:
So as to promote one's health; in a healthy way: If you eat healthy, you'll probably live longer.
The American Heritage Dictionary
I'm not sure why other dictionaries are so slow to add this definition. It is used more often than "healthily". When I searched COCA,
eat healthy _y* had 43 matches and
eat healthily _y* had 4. (Note that
_y* matches punctuation, to avoid matching healthy as an adjective. See this image for examples of what was matched in the
eat healthy _y* search).
There are many adjectives that are used as flat adverbs (i.e. without "-ly"), even when there is a non-flat form. The paper "Love Me Tender": A Usage Study on Adverbs Which Keep Their Adjectival Form examines this.
It's only some prescriptivists that view these types of adverbs incorrect. Apparently the reasoning was originally based off Latin grammar:
As Emily Brewster notes in a short video about flat adverbs, prescriptive grammarians in the 18th century, being overly attached to Latin grammar, thought flat adverbs were really adjectives being used incorrectly, and warned against their use. Before this, flat adverbs were more common and varied than they are now. Exceeding is a good example. If we browse Daniel Defoe's writing we find such phrases as: weak and exceeding thirsty; it rained exceeding hard; it is exceeding confused. Today this usage has an archaic feel.
Flat adverbs are exceeding fine
For an example of how opinions differ, Garner's Modern American Usage is fine with flat adverbs, saying that both "drive slow" and "drive slowly" are correct. Other resources, such as the Purdue OWL believe that it's incorrect to use "adjectives" this way (saying that "ate her lunch quick" is incorrect). If you're using a style guide, be sure to check out what it says.