If the crimes were committed entirely off base against non military victims, then probably civilian court, followed by additional UCMJ punishments and discharge.
I had a roommate who got drunk and assaulted a cop (it went very badly for him). He remained in the military for his surgery and court time while confined to quarters, reduction to e-1, and forfeiture of pay. He had a civilian trial and was convicted and served 90 days. Then came back to be discharged. Oddly, I don't even know what discharge he received -- he was my roommate when he went out that night, and wasn't after that.
> leaking sensitive customer call records stolen earlier this year from AT&T and Verizon
It says he was arrested for "two criminal counts of unlawful transfer of confidential phone records". This would seem to refer to "selling and leaking sensitive customer call records stolen earlier this year from AT&T and Verizon". No mention of arrest for the NSA hacking claims.
Not much info in the article about lots of things which could impact this.
First, it's in South Korea -- on base or off?
Personal computer or government furnished equipment?
What are the treaties in place for South Korea with regard to this?
What is the location of the victims' data for this?
Big things which could make this a UCMJ case would be (in my completely non-professional guess): He did it from somewhere on base. The victims were other military personnel. He used government equipment to do this. The US wants to assert jurisdiction according to a treaty.
All of this said, with the NSA issue hanging over it, there may be efforts to use the more easily proved crime to negotiate everything they can learn about the NSA claims.
tl;dnr - they can choose the jurisdiction. One side may assert priority and win (probably the feds).
The only thing I'm certain of is that every enlisted member will be brought in for a safety brief.
Which, notably, are Article II powers. Your usual Constitutional rights are abridged or suspended.
The USMJ limits some rights but also expands others. For example, a “jury of your peers” can more accurately reflect the accused, you receive Miranda rights earlier, you can request summary judgment for criminal matters, etc. On the flip side, free speech is more constrained due to the societal need for an impartial military, and this has been upheld by the Supreme Court on multiple occasions.