That's gonna depend on if the phone is GSM unlocked and if the NZ carriers support the frequency bands.
I would try to find out which bands are on your phone and then look at the carriers in NZ to see which bands they use.
If you use non new Zealand sim, you will be effectively roaming and you will get huge bill! I'd your device is unlocked, you can get a local sim. If not, there are ways to unlock it, your carrier can give you a code...