Due to my lack of vacation days, I can only squeeze out a 12 day trip, which inlcudes travel time. We would be leaving from NYC - do we have enough time to go to New Zealand? Since it's such a long trip, I don't know if it's worth it to go for such a short time (I am still bemoaning my lack of vacation days - when you have wanderlust like me, it's not a good combo!)
We are also thinking of going somewhere in Asia. We've flown to Thailand from MYC before so we're familiar with long plane rides to Asia.

Yes. I would recommend only going to the south island. Does the 12 days include travel time to/from NZ?

I would fly into Christchurch or Nelson (prob connecting through Auckland) and out of Dunedin or Queenstown (through Auckland).

My recommendation for any holiday down here is a minimum of three weeks if you want to make it worthwhile to see both islands. However if you only have 12 days I would fly into Christchurch and out of Auckland making a one way journey thru the country. If you hire a car drop it in Picton and pick another up in Wellington, don't take it on the ferry crossing, waste of money.
Forget Asia, for 12 days it would be ridiculous to put that in too. NY to here is a real hike, I know I did it last year in one go and never again.

For twelve days, stick to one island. The South Island has the iconic NZ scenery. If you arrive early in the morning, as many flights from the US do, it might be worthwhile to book a connecting flight the same day to the South Island, Queenstown is a good spot to start. Or Christchurch. The interisland flights are harder to get now that ANZ has a virtual monopoly. You can also fly nonstop Auckland-Dunedin.

Then on the return, fly out of the opposite end of the South Island.

I've been to NZ several times and I think you could have a great time there in only 10-12 days IF you limit yourself to one island. I also recommend that you choose the South Island for your stay. So go for it!

You might want to see my trip report. We were in NZ for 18 days, but 5-6 of them was visiting friends in Auckland. Flew to Queenstown and spend 12 days driving up back to Auckland. 9 days in SI, 3 in NI. Not enough, but doable.