One of my best friend's paternal ancestor, from whence her father's Maori family obtained her surname, was an American whaler.
However claiming the islands is a much harder goal. I could certainly see someone trying but they'd need to do it before1840 if they wanted to avoid a proper British claim. I would also think the British would be pretty annoyed as NZ was in the economic hinterland of NSW/Victoria at the time.
The fact that the UK, despite being the closest power with a large presence nearby, only sought to gain sovereignty in 1840 after half a century of being in the neighbourhood and then actual settlement/settler government not starting for another decade or so, speaks volumes about the desirability of the place to far off powers.