What is the difference between these two sentences: when will we eat? When are we going to eat?
If you're saying something is going to happen in the future based on present evidence, facts or will, then you use 'going to':
the sky is gray, it is going to rain.
If you want to talk about the future in a predicting manner, or you made a decision, you can use 'will':
you'll like her, or
I will not buy this record, it is scratched. But if you've already made a decision and you want to refer to it, then you use 'going to':
I think I'll have dinner with her. Yes, brother, I'm going out tonight.
Answering your original question and based on what I just wrote, you'd use
when will we eat? to appeal for a decision to be made, and you'd ask
when are we going to eat? suggesting that the decision of "when to eat" is already made and you just want to be informed of its outcome.
In everyday use, I dare to say native speakers would choose will to talk about spontaneous decisions, or predictions when no evidence is available. It is also used to make promises.
Going to, on the other hand, may better be used when talking about plans you've already made or when making predictions based on evidence (E.g. You have a look at the sky and notice the gray clouds and then say "it's going to rain").
Of course, language is a matter of use most times, and different users may have different perceptions on the "correct" use of certain structures.
Now, I almost forget to refer to your question. Both options might be correct in a given context, but if making plans is what you have in mind, I would say "going to" is the best option. If you ask sb "When are we going to eat?", you are supposed to agree upon a date, place, etc, and that's what planning is about. Hope it helps!