There hasn't been many details on the Camera besides the ones already listed, all we know it's a Tactical camera design to replicate the one some characters use in Ghosts. I'd like to believe there will be the ability to mount it to multiple surfaces so you might be in luck but don't take my word for it.
Comes with a curved surface mount, general surface mount, and a handlebar mount. The curved surface mount has loops so you can use the provided strap to attached it to a helmet or something. It also comes with permanent 3M double-sided adhesives to secure the mount if you want. Lastly it comes with an L-bracket. The camera has to be oriented in a specific way, with the record button facing up. So if you attach the mounts to the side of a surface (i.e. the side of your helmet), you can use the L-bracket to keep the camera in the recommended orientation. All in all, lots of options here.
Depending on the barrel of your paintball gun, the handlebar mount might work. The base of the handlebar mount is a slip-proof rubber and has velcro straps to further secure it in place. May add some weight though, not much, but more nonetheless. The camera (and Class 10 micro SD cards) are supposed to be shockproof, but not sure how much abuse it can take honestly.