Can the Army loosely organize into a union?

Knowing that you are "owned" once you are in military service, I am wondering if it is legal for soldiers who are up for reenlistment to loosely organize. Everyone knows you must obey orders, but the bane of an effective all- volunteer army is reenlistment. If an organized social network of soldiers who are up to reenlist were to mount a solid front, through lets say... social networking sites, which are relatively new to the scene, then theoretically it could yield leverage unavailable previously. The advent of social networking, coupled with the inherit weakness of an all-volunteer force which depends on reenlistment, well... you can imagine the power shift that possibly could entail. Has this ever been thought of before?