If you are a US citizen and would like to move to the Islands and it is not US owned territory do you have to seek citizenship from that country to stay aboard? And is that even possible to do if you aren't planning on marrying anyone in that country.

You don't need to seek citizenship of that country. As a matter of fact, it's not that easy... each country has different requirements.

Find out what you want to do and figure out category of visas you would need from employment visa, retirement visa or investor visa.