As far as I can tell, the springs there closed in 1974, and I can't see any accommodation still active in what is now basically an abandoned town. Is it legal to free camp there still, now that it's been reclaimed by the government?
Well here is the straight scoop from the guy who has lived here and run the place for 31 years. National Park Service regulations state that there is no road-side camping along Zzyzx Road, nor at our facility here at the California State University Desert Studies Center. Visitors are asked to park at the Park Service "Orientation Center" parking lot near the end of the pavement of Zzyzx Road. Walk-in access to the historic site is open 7/365 during daylight hours.
We are certainly not an abandoned ghost town as is often reported on the web. Far from it. We are a field station of the California State University (since 1976). In 1994, our site was placed within the Mojave National Preserve by an act of Congress, and which is now administered by the the National Park Service. Their regulations and rules currently apply. This land was formerly under the jurisdiction of the Bureau of Land Management. Wanna know more? Go to our website at: http://nsm.fullerton.edu/dsc/
The way I understand it is that Zzyzx (I love this name) is part of the Mojave Natural Preserve in which camping is allowed, if not encouraged seeing as it is one of the main leisure activities for visitors. Both camping at organised camp grounds, as well as roadside camping are allowed. To this purpose, the Mohave NP Camping regulations explicitly mention Zzyzx Road, stating that it is forbidden to camp within 1/4 mile of it:
Camping is not permitted: within 1/4 mile of any paved road or the Zzyzx Road;
It would therefore seem that you can camp close to Zzyzx. You can definitely pass by the settlement though since a self-guided trail begins there. Indeed Zzyzx Road becomes Paul Tiberius Road which connects to Old Government Road/Mojave Road, and the latter is one of the many trails through the Mojave NP.
After some more research myself, I found the Wikivoyage article on the Mojave desert states:
The best way to experience the Mojave is to camp. Much of the area is Bureau of Land Management land, with open camping allowed.
The Mojave Road Dry Lakes Dumont Dunes Kelso Dunes Zzyzx
So it appears that open camping is possible on BLM land, including the area of Zzyzx.