Amcam Macarca öğretmeni. → My uncle is a Hungarian teacher (i.e. a teacher of the Hungarian language). YES
Amcam Macar bir öğretmen. → My uncle is a Hungarian teacher (i.e. a teacher who happens to be Hungarian).
(when there is an adjective (Macar in this case) we don't omit "bir")
Just to clarify: The indefinite article is optional for nouns, i.e. "Amcam öğretmen" and "Amcam BİR öğretmen" can both mean "My uncle is A teacher", although the former can also mean "My Uncle is THE teacher", depending on the context. Since "Macarca öğretmeni" is a compound noun, it also follows this rule, so we could also say "Amcam Macarca BİR öğretmeni", but it isn't required. However, when there is an adjective we are required to used the indefinite article, e.g. "Amcam Macar BİR öğretmen". Is that right?
except that you cannot add "bir" in the middle of a compound noun; it should be "Amcam bir Macarca öğretmeni" (Amcam Macarca bir öğretmeni" is completely wrong)
Also "Amcam Macar öğretmen" might mean "My uncle is the Hungarian teacher", although I woudl rather say "Macar öğretmen benim amcam" for that :)
So I would call it "almost exactly" :). Anyway, sorry about that. :)
There are so many Hungarian references in this course. Are you guys planning a Türkçe-Macarca course anytime soon? That could be a really interesting endeavor. Maybe those friends and uncles could join the effort. :)