Nope, Cook Islands and Niue are definitely both countries. NZ citizens do not have the right to live in Cook Islands; if the Cooks were a territory of NZ then residence there would be an automatic right. I've flown from NZ to Cook Islands, and the whole customs/passport thing between the two is totally international. The Cooks and NZ maintain consulates in each others' (and other) countries and the Cook Islands has its own parliament and prime minister, and each country makes laws independently of each other.
I'm sure you're technically right but I think that on JetPunk a 'country' is considered one that appears in the 'Countries of the World' quiz. Maybe you could highlight them in red as you've often done in similar quizzes?