triple7nz,
I'm a foreigner that's been working as an instructor in California for the last 3 years. I started my path on J1 visa, where you start flight training, in my case IR, CPL, CFI and then work as instructor. Unfortunately, J1 for flight training programs no longer exists.
However, I am still doing what I am doing, now on an E2 visa. Don't think there is any luck for you there though. It's a "specialized skill" visa, in my case teaching towards JAA licenses. I already have my JAA fATPL and FI.
The school I am working still hires foreigners, though it mostly occurs internally (old students coming back as instructors). The latest thing seem to be F-visas. Not sure about the rights/limitations of these visas but I believe they all come on a FAA ATP training program where they train to CFI, CFII, MEI and then work as instructors in order to reach 1500 hrs. So perhaps applying to a school that has the same setup? You come here with the intention of getting an FAA ATP and will be able to work. Of course it's going to cost you, if you have a NZ license you'll probably have to get FAA IR, FAA CPL, CFI, CFII, MEI.
For the regional though, it's a dead end. Unless you get married or win the green card lottery. To end my story, I was engaged to an American girl and thought I had my life laid out in front of me, working as a pilot in the US, living in one of the best places on earth, San Diego. I was such a dreamer! Things changed for the worse when she suddenly passed away in a disease. Disregarding the emotional aspects (which were of course more important and saddening), I was back to square one with my E2 visa.
Now I just landed a job with a European lo-co so will be heading out of the US after 3,5 great years in SoCal. I have many American colleagues that just landed jobs with the regionals and comparing deals mine is better (salary, schedule, vacation), however the market is much bigger in the US and there are many more opportunities currently available so I can understand why people might be looking into US.