In the FAA Human resource policy manual(HRPM) under citizenship requirements EMP-1.13 it states (or it did if it is still in force)
"intial appointments to positions in the FAA made after April 30, 1998 will be restricted to USA citizens and nationals(residents of American Samoa and Guam) unless, as determined by the selecting official , there are insufficient number of well qualified applicants;or, for a brief period, in an emergency"
I looked into it several years ago and spoke to several managers in a certain region. As much as they wanted to take people on from outside the US , they all said it would open the flood gates. Plus there has always been a constant flow of ex-mil guys who fill the contract towers and then move on to the FAA.