On supermarket shelves with dozens of varieties of (sweet) biscuits, I find some packs (UK produced, inc. 'own brand') labelled 'cookies' but I cannot find any way (consistency, content) of distinguishing between them, and throughout my youth (I'm 75) they would have been called biscuits. In this link from a similar question http://www.pepperidgefarm.com/ProductLanding.aspx?catID=715, if the contents ('cookies') illustrated in the photos were simply laid out on a table, we Brits surely would call them biscuits. Forgetting 'dessert', we all know what 'eating a sweet' means and it would not include a piece broken off a bar of choc. Would an American say 'eating a candy'? Anyway, US 'candy' obviously has wider connotations.
To answer your question. No, a native American English speaker would not (normally) say "eating a candy", but would instead say "eating candy" or "eating a piece of candy". "Candy" is a mass noun.
As with many mass nouns, though, the plural, "candies", can be used to refer to categories of candy, and then "eating a candy" would be appropriate in a context where it means eating a specific type of candy.