Airlines have the right to make T&Cs regarding the use of certain equipment on board their a/c for whatever reason and the CC have the right to enforce these company rules if they so wish.
It is however utter nonsense to suggest that a camera will interfere with a/c systems. Commercial airline pilots by the hundreds, if not thousands, use their own personal cameras on the FD, including the filming of take-offs, approaches and landings (either from the jumpseat or by securing their camera to something. Furthermore, amateur and professional aviation video companies annually make hundreds of FD videos during all phases of commercial flights.