> If GPS satellites were to cryptographically sign their signals, would that be sufficient to prevent spoofing of GPS signals (assuming the receiver of the signals treated the signing appropriately)?
Nope, classic replay attack case (just record the GPS signal and replay to the device at the desired location). You'd need a true time signal within the device, e.g. and atomic clock to make it work (so you'd authenticate the signed time against true time).
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There is another way, however. If we assume the hardware is tamper-proof (unless drastically different methods are needed), then with strict timings we can device a challenge-response system that's immune to replay attacks due to relativity: simply transmit a signal A, have a known 3rd party (e.g. US government servers in cellphone towers) sign your signal Sig(A) and retransmit, and check that the delay matches the propagation delay you'd expect from the cellphone tower distance, plus the fixed (and immutable since it would be gov-controlled) processing delay. Your tamper-proof crypto-camera would record its location and whether it trusts the location. Using cellular signals is also better because GPS doesn't work indoors and is sensitive to interference.
Since we're adding a cellular connection to our device, it would also be a good idea to log its position on the state-controlled servers (again can be done with cryptographic safety assuming non-tampered device), along with some kind of intrusion detection system. As soon as it'd detect an attempt at tampering, it would relay this attempt to the servers, storing the intrusion and invalidating the authenticity subsequent recordings; a self-destruction attempt of the key would probably be also wise.
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And now that I think about it, you'd probably want to put several keys/auths in the device, from different organizations -- not only governments. That way if the government authentication is positive but NGO's mismatch, you can suspect a government-backed forgery attempt (analogously vice-versa).