Encourage New Legalized Immigrants to Start Training Act or the ENLIST Act This bill allows the enlistment in the Armed Forces of certain aliens who have been unlawfully present in the United States. Specifically, such a person is one who (1) was unlawfully present on December 31, 2012, and has been continuously present since that date, (2) was under 15 years of age on the date of initial entry, and (3) is otherwise eligible for original enlistment in a regular military service component. The bill also establishes a process by which the status of such an alien shall be adjusted to that of an alien lawfully admitted for permanent residence. It includes conditions under which that status may be rescinded upon a separation from the Armed Forces.