I don't think it's as implausible as some may think. "The USA hasn't shown interest in [x]" isn't an argument, since it is inherently rooted in confirmation bias. That is: the USA wasn't interest in [x] in OTL, for OTL-specific reasons. Change the specifics, and you change the outcome. We must as: was there a reason why the USA was never interested in gaining territory south of the equator? Some kind of aversion against it? Nope. It was just that no plausible candidates came up in OTL.
Distance isn't an argument, either. By that reasoning, Britain shouldn't be able to control New Zealand either. If it can be a British colony, it can also be a US territory-- and later on, a state (an evolution that I see going hand-in-hand with the development and larhe-scale use of reliable oceanic steam ships; the decreased travel time would make statehood so realistic as to become an inevitable desire).
Now, as to the premise: an earlier Civil War messes with the Mexican-American War, and thus with the USA gaining California-- which in turn severely limits the USA's ability to project any kind of naval power in the Pacific. Bad POD for getting this done, I'd say! Have an alternative: the nullification crisis escalates, South Carolina secedes. Andrew Jackson marches in, has things sorted out within a month, and in the process secedes Calhoun's head from his body. After this utter failure of a secession, the matter becomes a very embarrasing episode, and nobody sane will ever urge secession again. (And in fact, an amendment making it explicitly illegal is passed.)
The Mexican-American War goes through on schedule. With the issue of the South being very vocal about all sorts of things very decisively shut down for the time being, the USA is rather less tied up in the kinds of domestic troubles and compromising that marked this era in OTL. More attention can go to foreign affairs. Expansion was quite popular with comsiderable segments of the populace (and, somewhat more importantly, of the elite) in both North and South. But war over bits of Canada was considered insane by anyone... well, sane... and expansion into the Caribbean was too evidently a ploy to get the Southern slavocrats what they wanted (to the benefit of exactly no-one else).
So, New Zealand. That's actually not so crazy an idea. With the USA now having prime real estate on the Pacific coast, there will be interest in Pacific ventures. Initially of an economic nature, and then perhaps of a more geo-political bent. Just go with OTL as described in the OP's article, but with a USA that's not going to have a Civil War (at least not in the 1860s), and that has been somewhat more interested in foreign operations over the past two decades.
As a result, the OTL expectations just come true. Lots of americans settle in New Zealand, they soon form the majority, and they move for independence, followed by some kind of association with the USA. After this, even more Americans pour in, New Zealand is made a territory, and it retains both its large number of American inhabitants and its territorial status. About two decades later, it achieves statehood.