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Non-citizens are allowed to serve in the US military. It won't be a problem until he gets out of the military; unless, something turns up in his background investigation. If he gets out without citizenship papers and his legal residency term has expired, then he can be deported. He can fight it though if he has a clean background.
The bigger problem is that without citizenship, his opportunities in the military will be severely limited. He can't be granted a security clearance so he can't be an officer, warrant officer; or, in the top 2 NCO ranks. He can't serve in some types of units like Special Ops, Intelligence; or, EOD. The military's Qualitative Management Program (i.e. get promoted or get put out) may force him out before retirement when he gets to a point where he's not eligible for further promotions as a non-citizen.
Jump through the hoops. Better late than never.
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