Nothing unless it's close to somewhere strategically important to the US. But I'm not sure I see your point.

They want independence from the US, but they won't survive in open waters without the US warships. Otherwise, if this island has any value at all, it'll attract Mexican cartels or Russian mafia or whatever else is floating in the international waters these days.

A billionaire with no regulations could probably afford to hire enough protection to guard against anything but an attack by a state actor.

In any case, that's yet another reason we shouldn't allow American citizens to set up their own private countries.