Yes, I may have that wrong. At the time I was a contractor so the fear was the following situation per "If you're a New Zealand resident, you're taxed on your worldwide income."

If we wanted to eventually become residents of NZ, which we did, and I was obtaining income in the US, NZ and US would both tax me for that income.

It seemed, however, less likely that NZ would approve me for immigration if I was not intending to contribute my talents to a company within NZ, so in that case, if I obtained a job in NZ, I would be taxed in NZ, and US would again tax me as well for that income due to US citizenship.