Illegal immigrants are NOT subject to any jurisdiction. As proof, order them to report to the nearest police station or city clerk's office. What, no stampede?
There are lots of citizens who would decline that "invitation" (or reject that order) as well. They'll just send out armed men to bring 'em in, same for an illegal.
I think the key phrase is "born or naturalized in the United States", and the illegals are neither. Their kids born here are citizens through our own laws, until we get the guts to change them.