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.
Illegal is the wrong word for it. It's not illegal, it's a violation of the Facebook TOS - a matter for at most a civil (and not criminal) court.
That's not relevant, though, because US v. Lori Drew[2] decided that a user can't be prosecuted under the CFAA for breaking a ToS agreement. (BTW, IANAL.)
0: http://www.law.cornell.edu/uscode/text/18/1030
1: https://en.wikipedia.org/wiki/Computer_Fraud_and_Abuse_Act
2: https://wikispaces.psu.edu/display/IST432TEAM24/United+State...