The only rule for the US is that US citizens must use their US passports to enter the country. This applies even if you are a dual national holding a US and additional passports.
If you are US citizen, but don't have a US passport - yet hold dual nationalities, in this case you cannot be denied entry, but you will be subject to additional screening.
All other nationals can use whatever travel document is legally issued to them, assuming its valid for the type and nature of the trip.
In your case the only advantage of your UK passport is that its tied with previous successful entries (ie, it gives you a good travel history of not violating your visa terms).
Assuming you have never traveled on your NZ passport to the US; this may cause slight delay as the first time around everyone is vetted a bit more closely - especially if you are at high risk of violating your visa terms (a student, for example).
However, once you have established a solid travel history on the NZ passport then you may use whatever is handy at the time.