"We are not eating lunch."
Translation:Wij eten geen lunch.
If you negate something, you're negating it according to the rules of a given language: if in a situation where you'd use "don't" in English you're now forced to use "geen", they mean exactly the same because they carry the overall meaning of "sentence or word used to negate something" and as such they're synonymous.
It's precisely because "We are eating no lunch" sounds off that you use "We're not eating lunch"... Remember that Dutch speakers don't really have a choice here, so this is their most natural way of negating a clause - as such, you can translate it to the most natural way of negating something to you, an English speaker.
P.S.: You can translate geen as "no [something]" when that makes sense/seems right in English:
- Dat is geen koe. Het is een paard. (That is no cow. It is a horse or That isn't a cow...)