If you are in legal status as a nonimmigrant, or a beneficiary of TPS, you may be eligible to obtain citizenship “by the time [you] graduate from ten weeks of Basic Training or commission” in the US Army. Compare that with the time it takes to get a green card, plus the three to five year wait after the green card before you can apply for naturalization.
The US Army is only one branch of the military that has an ongoing MAVNI (Military Accessions Vital to the National Interest) program. Their MAVNI Information Sheet (shown below) describes MAVNI eligibility and qualification criteria. As of October 2012, it appears that the Army is quite pleased with the recruits it is able to obtain through MAVNI. Contact other branches of the military for their criteria. See also Julia Preston, Pentagon Reopens Program Allowing Immigrants With Special Skills to Enlist, New York Times (Oct. 27. 2012).