My gut reaction is to say no they should not. However, if hypothetically there were some question that came up that could only be resolved by checking their fac

My gut reaction is to say no they should not. However, if hypothetically there were some question that came up that could only be resolved by checking their facebook account, then I think it can be allowed in that instance. The difference is there being (some equivalent of) "probable cause" to even question something that could be resolved through such a check.