"She is eating chicken."
Translation:Sie isst Hähnchen.
We use an article if she is "a" chicken as opposed to eating chicken.
I suppose it would be hard to tell whether she is a chicken or she is eating an entire chicken. But since "she is eating a chicken" isn't common in English, I wouldn't expect it to be common in German.
If a German is speaking to you and you aren't sure whether you are hearing "Sie ist ein Huhn" or "Sie isst ein Huhn" you should ask for clarification. Or perhaps he might say "Sie frisst ein Huhn" since she is so voracious.