No it doesn't. It has jurisdiction over the rights of American civilians on American soil. Anything else is significantly more murky (e.g. Guantanamo Bay which is American soil but not under US civil jurisdiction).
If I understand the Wikipedia article correctly (and of course, if the article itself is correct) then the situation is actually the opposite: it's Cuban soil but under American jurisdiction.
As far as I can tell (and the whole situation is bizarre and confusing, so it's hard to be sure of anything), the prisoners there are under US civil jurisdiction, and the ongoing troubles they have with basic rights are due to the fact that they're "enemy combatants" than the fact that they're in Cuba. Being in another country seems to be more about making it in practice harder for US civil courts to exercise their jurisdiction, but not shut them out altogether. There have been quite a few relevant rulings by civil courts.
No, it is well-established that U.S. courts have jurisdiction over the rights of everyone on American soil. Guantanamo was contested because it's not clearly American soil (an American military base leased from Cuba).
The context is a discussion of Al Awalki, a US citizen not present on American soil. However, I'm happy to re-ask my question making that context more explicit:
Is the executive then to get free reign, unrestrained by the judiciary, to declare any given US citizen currently not on American soil the proper subject of an extrajudicial killing?
Is the executive then to get free reign, unrestrained by the judiciary, to declare a citizen an "outsider"?