Being a part of the world implies having embassies, which conflicts with "no one from a different country will be allowed". Embassy staff will, without doubt, consist of at least 50% spies and saboteurs. Probably more like 80%. There's a very significant amount of illicit stuff going on with them even today, in a no-special world.
The first thing to happen will be, witout any doubt, cowardish, deceitful acts of piracy with invasion attempts. In other words, the righteous and fair will do what they call "black operation". After realizing that none of the men who have no name and whom nobody sent return, and after denying any involvement, the respective nations will resort to the next step, diplomacy and treason. They might, however, start both strategies in parallel, both to save time, and for synergy.
Particularly malicious nations such as the USA will likely try to kill the population with biochemical warfare, and blame it on the Chinese. Oh wait, that is already happening right now. Forget what I said. Russia might use a neurotoxin, and later reply "So what? Prove it. I see no evidence." when challenged. Oh wait, that has already happened, too.
The most likely thing is they'll one way or the other try to assasinate the government and key persons (and temporarily disable defense infrastructure), and take over the otherwise completely functional island with the mostly helpless but knowledgeable civil population. To quote Guzman (or Axl Rose, if you prefer):
We practice selective annihilation of mayors and government officials, for example, to create a vacuum, then we fill that vacuum.
Other than that, if this fails, there will be no significant change.
As you said, the isle will not share their technology with outsiders, so they are not interesting for the rest of the world. At best they are a threat, so other nations will be somewhat bootlicking them, just in case. Nobody will try to start an open war.
Although, they would likely not be bootlicking too much as it is obvious that they will not be a "powerful ally" in some prospective war. Why would they, there's no reason for them to give such a guarantee or even engage in a war that isn't theirs. Although history tends to repeat itself, and the stupid never die out, one should nevertheless assume that someone so advanced should be aware of what happened only a hundred years ago. They'll not repeat that mistake.
Also, "respond violently" is very likely to be seen as a relative term. If they are sufficiently advanced as they are, they will certainly not start a war by themselves because they will know that they will lose either way.
War means, at best, to lose a couple of drones and drive forward global pollution, and at worst (or rather, in the to-be-expected case) ending up with an awesome island that has fusion power and energy shields, but is located in an ocean with, uh, radiant water, so to say. And the rain is black, and skies are grey.
Shields may be able to block a blast, and drones may be able to destroy every ship and airplane. You may be able to speak the word, and have fire and brimstone rain from the skies onto the heathens, whatever.
Be that as it may, you are still dependent on the rain that nature kindly gives to you, and you are still surrounded by an ocean. You still need to breathe, too. If the rain is toxic and the ocean is radioactive, that's not a lot of fun (unless you also have gate technology and have already terraformed a Juiter moon or something, then again why would you stay on a small isle...). You still need the sun, both to grow plants, and to not go crazy. So... when the sky is clad in one huge dust cloud for a decade or two, where's the sun?
So... starting a war is very likely not what you'll want to do, even if you know that you will win. It will be a Punic victory (which I think isn't a real idiom in English, but very well in other languages, it means as much as lose-while-you-win). Jean Jaurès, the only intelligent socialist in history, was wise enough to know that, and courageous enough to say it in public. Unluckily, a bullet to his head from behind was the only reward he got from doing so. No good deed goes unpunished, heh. But we should assume that a very advanced society is a bit smarter than early 20th century French nationalists, so they will know that going to war is generally a bad idea.
On the other hand, the rest of the world is not very interesting to insulars either because the world has nothing to offer that the insulars cannot make themselves (except Korean girls). So... interaction will be rather limited, there's not much of a base for trade and the like.
Seeing how the island is close to the west coast of Africa, though, two things are going to happen for certain. Traffickers will get super super rich from selling places in low quality boats, and you will have thousands of dead bodies being washed upon your shores every week (unless you rescue them, which you won't, as this would be letting other people in).