I'm certain that anyone asking to snoop through your Facbook knows they are doing something "wrong" - maybe they don't understand the full implications, but I'm sure none of them would be stupid enough to put the request in writing.
In many US states, it is legal to record a conversation if one of the parties is aware (i.e. if you are recording a conversation in which you are a participant).
I'm uncomfortable with doing this in normal circumstances, since it's a pretty manipulative way of getting a recording of someone. But if you're getting abusive requests from your employer and (I agree) you're not likely to get it in writing, this seems like the appropriate response.