The world has an inexhaustible supply of young, military-aged young men anxious to better themselves by emigrating to a first-world country without permission. Meanwhile the young men of those first world countries seem to have things they'd rather do than be soldiers and sailors.
This suggests the confluence of a problem with a solution. As an alternative to deportation, how about offering the healthy illegal aliens enlistment in a foreign legion with a US officer corps? What follows is very much a back-of-the-envelope first draft.
The offer: room and board, uniforms, medical/dental care, ESL, a modest monthly stipend to cover incidentals at the PX (insufficient for drugs/booze/whores/cars), and a deferred salary held hostage for their continued good behavior, forfeited if they screw up.
Promotions available up to senior sergeant ranks, based on performance and good behavior. Enlisted personnel must live on base - unmarried - for their first four year hitch. At the end of which they can take their accumulated pay and go home. That lump of pay should go a long way to establishing a life in a third world home country.
Or they may reenlist and marry if they choose, live on or off base, and draw their pay like any enlistee. Enlistees retain citizenship in their home country prior to retirement from the legion. At the end of 20 years of honorable service, retire with US citizenship (if desired) and live where they choose, full veterans benefits only available in the US.
Prior to retirement, all legionnaires are subject to military discipline. Gross misbehavior anywhere along the line means demobilization and deportation, with forfeiture of benefits.
Obviously there would be hundreds of details to be worked out concerning travel documents, legal status of home country wives, etc. This outline is the result of maybe an hour's cogitation.