I am somewhat at a loss as to a particular hike to recommend.
This is because hiking in the mountains, in winter, overnight, is really better to do if you have some experience, and know exactly what you are doing.
The thing is, severe winter conditions can appear, in minutes, with little to no warning. While this can happen at any time of the year, winter is by far the worst, and gales, snow, and storms arrive from the south rapidly.
You also need to be able to carry around thirty pounds of gear on your backs, and that is not easy, as food, sleeping bags, dry clothing, gas stove, etcetera, add up very easily. You need to bring proper tramping (hiking) packs, as you cannot depend on hiring them here - although it is possible, and in any case, it is best to have your own pack, adjusted for your own back.
It is not unknown to need gaiters, crampons, and other gear for wet and snowy weather, even if you are sensible and remain below the bushline (generally around 1200m/3658ft, in the North Island, although this, of course, varies.) Aside from the hazards of the tramp itself, I am wondering if it is viable to bring all this gear with you.
My recommendation would be that you choose either Tongariro or Egmont National Parks, or perhaps Te Urewera if you are heading to the East Coast, all on the North Island, get advice from the respective visitor centres in those National Parks, and make a decision which hike to do on the day, and do it as a day tramp. You can visit a hut that way, and I think it would be a safer way to do things.
If you are determined to do an overnight stay, you need to buy hut passes - $5 or $15, at the visitor centre, depending on the services provided. Overnight stays are not guaranteed, and operate on a first-come, first-served, basis.
Most huts provide a wood stove, a supply of firewood, a rainwater tank, sinks with no taps, a pit toilet, and platforms with mattresses. Comforts are very few, unless you are in the Great Walks Huts, which cost rather more. Water from the tank is generally safe to drink, but boil water taken from mountain streams due to the risk of giardia. I sometimes do drink from mountain streams, but only once I am above the bushline, and amongst the tussock. That is because I know the area, don't take a gas stove on day hikes, and am comfortable with the risk. Below the bushline, I don't do it.
I am very sorry to be so discouraging, but I really think that if you are not experienced with winter conditions - severe or otherwise, it would be better not to head out overnight in the winter. You have to be prepared for anything and everything, carry a fairly comprehensive first aid kit, a locator beacon is a good idea too, and I am also wondering about the logistics of carting in all the things you need from the United States. You also may get stuck due to the weather, and this can cause much anxiety and agitation if you need to be somewhere else at that time, or worse again, have an aeroplane to catch the next day!
Now - recommended tracks. Stick to the walking or easy tramping graded tracks - tramping tracks are often guided by cairns or markers, and it is easy to get very lost, if bad weather arrives.
At Tongariro National Park - try the Whakapapaiti Track (18km, has a hut), part of it is on open tussock, the rest is in the bush. I am unsure for other recommendations, as the other tracks I looked through for you have rather more tussock, and it gets very exposed when you are that high up. Tukapakurua Falls may be suitable, although there is no hut. Otherwise I would suggest short walks, like the two hours to Taranaki Falls, if the weather is good. Tama Lakes would be a good walk if it wasn't winter - even in the summer it can get very exposed.
For Uruwera National Park, the Lake Waikaremoana Track is part of the country's great walks - which generally means several days of tramping, in this case 3-4. As they are Great Walk Huts en route, they cost $32 a night. Lake Waikareiti is a two hour walk - beyond the day shelter the track becomes rather more challenging around to Sandy Bay Hut, and I would not suggest you do that in winter.
For Egmont National Park, I am on firmer ground, as this is my home soil.
Maketawa Hut - follow the Puffer - aptly named - from the North Egmont Visitor Centre, then turn off to the Maketawa track. This walk involves some short ladders up steep banks - at least, it did, the last time I passed that way. You can return the same way, or there is an alternate route through the bush - you will need the skill to cross unbridged streams though.
The Holly Hut Track is definitely not suitable in winter - very exposed areas en route, and you need crampons and an ice axe available.
Waingongoro Track heads south-west from a bit below The Mountain House. Again, you need the skill to cross unbridged streams, and if they are swollen with heavy rain, don't cross them. There is a hut en route.
If you use the LOWER Lake Dive Track, departing from Dawson Falls, it may be suitable, as the higher track, again, needs an ice axe and crampons during the winter. It is 6 - 8 hours return, to the Lake Dive Hut, from the Dawson Falls Visitor Centre.
All of the areas I have described have multiple short walks of 2 - 3 hours, and these are likely to be more suitable at this time of year. I hope this helps you to make a decision.