Lit, an American citizen must leave with his American passport. I presume that there is a computer match to the Social Security number. Snowden left before being discovered. As for Israel, extraditing residents to the US, that door swings both ways (although Israel has conditions like no death penalty and right to serve sentence in Israel). The Tzitz Eliezer discusses the halachic rationale for allowing it in Torat haMedina (while Rav Shaul Yisraeli disagreed it could be that the service of sentence in Israel clause did not exist then).
Joseph, if they are not at least legal residents they all have to leave periodically when their visas expire.
Akuperma, that is a spy attitude.