There is absolutely nothing wrong with "What are you eating?" In fact, I almost prefer it to "What are you having?" (which is also fine).
However, "Eat the dinner" sounds unnatural. Are you sure she didn't say "Eat your dinner"?
We use "to eat" in many different ways. Here are just a few random examples:
I was eating lunch when my house fell down.
What do you want to eat tonight?
Where are we eating on Saturday evening?
I don't eat peas.
Eat up before it gets cold.
The guilt is eating me alive. (Metaphorical use)
I haven't eaten for days.
He died from eating bad mussels.





