If the product is created in NZ that uses technology patented in US for example, would it still be legal for them to do business in US, like offering services?
That depends on who is suing you, and where they file.
I remember a decision from a few years ago that said something to the effect that any activity that involved transmission of data through the US could be interpreted as having happened in the US, even if the computers, their owners, and operators all resided outside the US.
Practically speaking I doubt the typical east Texas troll even has a clue what kind of shakedown letter to send abroad.