Pitcairn Island gets its support from New Zealand, so NZ calls the shots and pays the bill. Easter Island has no long-term tenable anchorages anyway. French Polynesia gets its nurses from France on 6-month or one-year contracts. Speaking French is important because the locals speak French as a second language, few speak English outside of Tahiti. And I doubt that you could get a work visa. If you were interested, you would have to inquire before you got there.
French Polynesia requires that foreign yachts leave French Polynesia by November 1, and they are quite strict about that. They will not allow the crew of foreign yachts to stay through cyclone season. You can leave the boat on the hard, but the crew must leave.
American Samoa is a US protectorate, and US citizens can work there without work visas or anything. Lots of Americans crossing the Pacific take a year or longer rest in Pago Pago to work and build up their cruising kitty. Doctors and nurses tend to teach at the community college - definitely worth inquiring into.
I have less information about the next few island groups, so hope that others can fill in the rest of the Pacific crossing. Guam, though is always another option for working, though it's a long way to go for most yachts crossing the Pacific.