"He has no brothers/siblings (if you mean both borthers and sisters); neither have I".
"He has no brothers/siblings. Neither have I".
"He has no brothers/siblings and neither have I".
I believe any of these are acceptable.
PS: Note that you could also say "Neither he nor I have any brothers/siblings" or "Neither of us have any brothers/siblings" with the same meaning.
Student or Learner