I wouldn't have belittled your original wording, although Bill's done an admirable job of explaining why it might sound a little bit off. (Not wrong, just a little off, if that makes sense – and I'm not so sure that's even true).
I checked online to see if I could find some authoritative source that might shed light on this, but I came away even more confused than when I started. Maybe I should've just stuck with Bill's answer.
I did find this at one website:
When using "neither" in a sentence, you are saying not the first object and not the second object are behaving in a certain way. For example:
Neither Corie nor Bob went to the play. (Corie isn't going to the play. Bob isn't going to the play.)
That made me think, "Maybe this shows which side of the verb neither/nor are supposed to be on. Maybe it's okay to say "Neither X nor Y have Z", but not "Z has neither X nor Y." But then my next source contradicted that theory:
A “nor” usually follows a “neither” when they're used in the same sentence. For example, you might say, “I like neither hot dogs nor mustard.”
This made me wonder: if the Grammar Girl can say, "I like neither hot dogs nor mustard," then why can't Crazy Buddy say, "I have neither a brother nor a sister."
In short, although I wouldn't say that "I have neither a brother nor a sister" is a very natural way to say it, I can't find anything wrong with it grammatically. Moreover, as I said in my comments, I don't find "I have no siblings" to be dramatically better than your original wording. While I might agree with your teacher that "I have no siblings" is a better way, I can't agree that it's the right way, while yours is a wrong way.