PhD in New Zealand and Post-doc in US
Yes of course you can move from a PhD in NZ to a postdoc in the USA if you publish good work in good conferences/journals. You could potentially study for part of your NZ PhD in the US on a scholarship such as the Fulbright. If you have a PhD offer in NZ then presumably you have some idea of what subject you'd be researching - does that field excite you? Your decision should be primarily motivated by interest in a particular field of research rather than by country (IMO). 

A PhD in NZ/Australia/UK is much shorter than the US (3-4 years compared to 5-7 years in my field at least) and generally doesn't involve coursework - you'd also be adding an extra year on top of the US timeline to sit the GRE exam. One final comment: that you have a PhD offer from NZ and are considering waiting a year to sit the GRE and do a PhD in US makes me think you aren't that interested in taking the PhD offer in NZ - there are few things in life I would discourage as much as doing a PhD without a passion for the subject.