As I usually do in places like this I will advise you should have an suv or truck and 4wd before even bothering with this area.
It’s remote as can be and there isn’t a single thing out here including cell service
As far as the camp site well it’s about a 40mi trek through the empty desert to even get to the bridge and along the way there are numerous areas to get your rig pulled off to the side and camp it out however not every one is perfect so just do some hunting for the one that suits you
The bridge itself is pretty cool and apparently there are hot springs somewhere around here from what I’ve read but I didn’t search for them or see any signs. There are also some old structures but be respectful and keep ‘em protected
No water until the verde river so keep that in mind. Cell service is non existent as well. There is shade at spots due to the local fauna but for the most part it’s barren and you need to prepare for the trip out here wisely. Nearest entrance is off 17 as well as amenities so stock up and have fun!