I also think it would be very difficult to become an ant.
While it's not particularly challenging to visit the USA for a short period of time, or even a few months with evidence of "a plan" as a tourist, I can't imagine the border and immigration people would be very used to dealing with a request such as yours would be. If they haven't heard about it before, it's suspicious.
There would also be possible repercussions and expenses to you and your family upon your return to New Zealand after a lengthy time away.
For example, as a Canadian, if I spend more then three months away from Canada at one time, I have to pay higher travel health insurance, and after six or seven months, I would lose my access to my provincial public health system, until I came back and spent three months in residence without any coverage for that length of time. There would also be tax complications.
But after six months, without a visa, I would be kicked out of the USA anyway. I don't know what length of time a tourist visa (if you would need a tourist visa) from NZ would allow you to remain in the USA.
I really think it would be easier if you thought of arranging as a gapper, because unless you really do want to move to the USA for the rest of your life, it would probably be very stressful to arrange.
I would just apply (or show up, again, I don't know how the NZ/US travel works) as a normal short term visitor, learn what you could, and take it back home.
P.S. - I don't think the USA is particularly difficult about visitors from other countries (YMMV) after all - what would New Zealand say if someone from the US arrived wanting to be a unpaid labourer on some sort of communal farm
project for a year?