EA owns everything when it comes to the SS license. They own the IP and the distribution rights. All of it.
EA owns part of the rights, but an insurance company called Meadowbrook owns the other half.
Back in the day, Warren Spector split the rights between EA and Looking Glass so both of them would have to cooperate to make any future games in the series. When LG went out of business, those rights defaulted to an insurance company. Here's a quotation from the man himself when he was asked about a potential Shock 3:
"Unfortunately, the rights to System Shock trademark and copyright are both up in the air. I feel so guilty I actually negotiated the deal that said Electronic Arts owns the trademark and Looking Glass owns the copyright. My thinking was it would force us to be married so it never would be that either party should be able to say we own that, we’re making the next game, screw you. What happened was, when Looking Glass went out of business some insurance company or something ended up owning the copyright to SystemShock while EA still owned the trademark and so now, I literally don’t think anyone knows who owns what rights anymore. But if anyone is going to do it, it’s probably going to be EA and I don’t work for EA. The rights are very confused."
Here are some links so you can read more. http://blogs.mercurynews.com/aei/2010/11/28/warren-spector-interview-on-the-process-of-making-epic-mickey-system-shock-and-dark-games-part-2
So no, EA does not own "everything" when it comes to System Shock, and this is why those games are not on GoG, I'm sure EA and GoG would love to release those games, but it's going to take some serious legal kung-fu.
Educated and researched. Thanks. Although, by reading the 2nd article, it does seem that EA could indeed resurrect the franchise if they wanted. I'm sure the EA lawyers could make magic happen if they really wanted it to. Legal mumbo jumbo is all in a days work over at EA.