It depends on what you are responding to and whether it is one thing or two things that you don't have. If someone said, "I don't have a dog," you could respond, "I don't have one either," or, "Neither do I." If someone said, "I have a dog and a cat," you could say, "I don't have either," or, "I have neither."

No - the order of the second is unnatural. It's yoda-speak. It would be better as 'I have neither'.