Well, you didn't really describe your situation very well in your opening question telling where exactly you are at the moment.
When you land at the airport and arrive at the immigration queues, there should be an officer there guiding people which queue to join (foreign passport holders queue or New Zealand citizens/residents queue). You should tell that officer that you are a dual NZ/USA citizen, but you do not have a valid NZ passport. They would probably take you aside and verify your details and land you. If you don't see an officer before the queues, find one you can talk to before you get into any queues.