Encourage New Legalized Immigrants to Start Training Act or the ENLIST Act This bill authorizes the enlistment in the Armed Forces of aliens unlawfully present in the United States on December 31, 2012, who: (1) have been continuously present in the United States since such date; (2) were younger than 15 years of age when they initially entered the United States; and (3) are otherwise eligible for original enlistment in a regular component of the Army, Navy, Air Force, Marine Corps, or Coast Guard. The Department of Homeland Security shall adjust the status of an alien enlisted under such authority to the status of an alien lawfully admitted for permanent residence under provisions of the Immigration and Nationality Act applicable to aliens who entered the United States prior to January 1, 1972. The bill rescinds such lawful permanent resident status if the alien is separated from the Armed Forces under other than honorable conditions before serving the term of enlistment.