Surveillance cameras and smart phones are everywhere. You have to assume that everything you do can be recorded. If you do everything right, it would seem to be beneficial. But, like in football replay reviews, you can be second-guessed?
Can you be criticized for the speed you do something, too fast or too slow, especially EMS work? Can an attorney pull a small fragment of a hostile situation out-of-sequence and make you look at fault for restraining a difficult civilian you are trying to help? Can something you say, which is not what you meant to say, or taken out of context, be recorded and intrepreted negatively against you? Answers are yes.
You can protect yourself at work by being well trained, follow procedures, don't take short cuts, work as a team and - maybe have personal insurance.
Even off duty, I can understand the chief's warning. You can get tickets for speeding on empty highways which have cameras and speed detectors installed. You can be towed from a busy airport parking lot which uses cameras to pick up over-24 hour parking violations. Anyone can be taking our picture or recoring our actions - anywhere, any time. If you are a little league coach these days, be careful of everything you say and do. If you have ice or snow on your home's walkway, remove it or someone may record it and blame you for a fall.
You have to be careful.