Saint Martin, Guadeloupe and Martinique are full EU, thus you can work there. Sint Maarten is not EU and, from what I've heard, even Dutch nationals need permission to work on that side of the island.
Reunion, French-Guyana, Canaries, Ceuta, Melilla, Azores, Madeira and Gibraltar have full working rights for EU.
Apart from that the rest of the overseas territories are not directly part of the EU and thus can impose their own work regulations. That includes all of the Netherlands Antilles(Aruba, Bonaire, Curacao, Saba, St. Eustatius in addition to Sint Maarten), the islands belonging to the British Commonwealth (BVI, Anguilla, Montserrat, Cayman, Turks & Caicos), the French dependencies (French Polynesia, New Caledonia, Wallis & Fortuna, etc.) all make their own work rules, and to be fair their citizens also do not have any reciprocal guaranteed work rights in the EU, either.
The problem is that the type of work that pays the boat-bills will most certainly be well-staffed by locals and the types of specialized work that pays well will not be found in those locations.
Saint Marten is a French Departement, you can work there. Sint Maarten is Dutch but not part of the EU thus you cannot work there as a citizen of the EU without a visa (I believe that even as a Dutch national you need to get permission of some sort). You can work in Guadeloupe and Martinique. I think that the Dutch Antilles countries are also not EU and thus a visa is necessary.
Addition
The definitive list of countries that are full signatories and thus where you have a right to work without a visa is on the pages of
EUROPA – The official website of the European Union but the clickable map is at
EUROPA - European Countries
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Working in US territories is probably not much of a problem, as you're a US citizen, but the rest of the Caribbean will be problematic.
