I love the Trona Pinnacles, and it's a cool place to camp. I'll have to pass on this trip because I'm leaving the following Thursday for the Rogue Overland winter expedition in Arizona, and wouldn't be able to do back to back weekends like that. However, I'll tell you now that it would be a great place to post up, even if it's just you and the Mrs. and EJ. It's not really the kind of place that can "close down," as there are no gates or anything. The worst thing that would be impacted by the government shutdown is that the bathrooms might not have been visited by BLM maintenance personnel, so just bring your own toilet paper (they're pit toilets with no running water).
There are usually a few small groups out there camping, but everyone intersperses within the pinnacles themselves so you're afforded some privacy with the security of knowing that others are around if you have an emergency or something. It is the high desert, and it does get chilly up there in the winter, so just be prepared for that with proper sleeping bags and clothing. The average overnight low temperature in January is 30 degrees.
