Is this an official policy? Why is nuclear such a big deal, and how does the military factor in? Is it just that those are the major things the US would like to sell to NZ?
* Anti-Nuke Background
The watershed was the 1985 sinking of the Rainbow Warrior, a Greenpeace protest ship, in New Zealand by French secret agents that lead to the death of one of the crew.[http://en.wikipedia.org/wiki/Sinking_of_the_Rainbow_Warrior]
This enraged the general public in NZ and lead to wide spread protest. Since the mid-1980s the anti-nuke stance has become part of the Kiwi psyche to the point where no Government would dare change it.
* Being anti-nukes effects NZ and US military relations
This causes problems with NZ and US relations because the US has a nuclear powered navy with nuclear weapons (ships and subs) and therefore no US watercraft are allowed in NZ territorial boarders as the US Navy has a "neither confirm nor deny" policy when it comes to the nuclear capability of a particular ship.
Also because NZ has a very small military budget, NZ relies heavily on the assumption that the US provides regional protection in the Pacific. This has lead to a strained military relationship. If you are interested in a summary read the Wikipedia ANZUS article.[http://en.wikipedia.org/wiki/ANZUS]
* Strained military relations lead to strained economic relations
The trouble that NZ has is that the economic benefits of free trade with the US are worth a lot more to NZ than the US. NZ really has no leverage except to let the US do what it wants, and the US cares deeply about unconstrained military options. Because of the constraint on the US Navy by NZ's anti-nuke stance, this means that until NZ changes, the US will not bargain with NZ on other issues like trade.
Yep, it's the law. Much to the chagrin of the US.
Their Military has no Nuclear weapons, and the population is pretty strongly opposed to anything nuclear, instead favouring 'green' technology.
I'd go as far to say, as it's now so entrenched in NZ's identity, if any government that tried to repeal it, it would be an act of political suicide.