You must lunch with the girl is the equivalent to "you must eat lunch with the girl"
You must lunch with the girl is also correct.
How exactly do we differentiate between "having breakfast" and "having lunch" in this kind of context?
breakfast is petit-déjeuner
That is a regional thing.
In Québec for example, breakfast is indeed déjeuner, and lunch is dîner (I will admit that the Québec one seems strange).