St. Maarten isn't part of the EU, but St. Martin is. But inhabitants of St. Maarten are EU citizens, and there is no hard border between the two parts even though there should be one, as this is a border of the Schengen Area.
This means that I, a citizen of the country of The Netherlands with a identity card but without a passport, am allowed to travel directly to France via Belgium, and I can book a flight and enter St. Martin, but I am not allowed to visit St. Maarten!
In France we differentiate between parts of France proper (mainalnd France, Corsica, Guiana), our overseas departments (used to be more complicated but now they're "almost" parts of France), and our overseas collectivity like St. Martin. But as a French citizen I can visit pretty much any that I want without the need to ask anyone for permission.
Additionnaly, how is "your" part of the Island governed; fully autonomous or has the royal family (real/actual) oversight or is the governement of the Netherland overseeing it or ... ? Our way is mostly "they have their local governement but if the French one tells them something they do it".
It's interesting to me, especially since New Caledonia being the only dependance who didn't ask to be integrated but to move away will have to decide between autonomy or independance
The Netherlands (the country) also has overseas territories, namely Bonaire, Saba, and St. Eustatius. They are considered regular municipalities, but you still need to have a passport to visit them. They also have some duration limitations to prevent migration from the mainland.
About the governance: Long story short, it's complicated. Some laws apply to the Kingdom so they're for both continental and overseas, but most laws are made only for country of The Netherlands. The other countries are mostly self-governing, including making their own laws, but stuff like international treaties and military is done on the Kingdom level, which in practice means it's done by the Country. To give an example: after Irma, St. Maarten had to formally ask The Netherlands for help, otherwise the military would not have been allowed to give any assistance, as it could be considered an invasion. In general, any form of intervention is seen as a form of neo-colonialism. We try to intervene as little as possible.
So France the (geo)political entity and France the country absolutely has a border with Brasil, and it is its largest land border.