"We are not poor."

Translation: Nem vagyunk szegények.

[wyqtor]

Nem szegények vagyunk - Would this word order work?

[Findil]

Yes, it would.

The only difference is that it negates the adjective directly. You could continue it with another clause that states what you are instead: "Nem szegények vagyunk, hanem gazdagok"

[AndresCollinucci]

So in this case dropping the verb to be is not acceptable?

[mizinamo]

Yes, because it's not van or vannak.

It's not "to be" in general that's droppable, just those third-person present-tense forms.