I suppose it's rarer where I live than in the U.S. midwest to have your boss's wife (or some other member of his/her close family) as a day-to-day floor workmate. Most people who landed in that situation and who are holding down a steady job with some responsibilities and a decent wage, not just a brief extra stint, would likely take care to join the union or set some sort of framework for their employment, albeit if they were allowed. Precisely in order to have some backing if there's fallout, backstabbing or favouritism coming out of the boss's family connections.
The boss may have been technically entitled to fire her, but the reasons he gave are absurd. It's like Darkling pointed out., she had not even done anything unseemly or threatening; the initializing of the "hot conversations" and sms rounds seems to have been all on his part, and that basically makes him into someone who has admitted he is one step away from becoming a stalker or sex offender.
If you can fire for this, you can pretty much fire for anything. Suppose the dentist and his family had been fervent believers in that the world would come to an end this week, as of the Maya calendar, and the dentist's dauighter had teased the dentist's son at school a bit about it. The boy came home and whined about it, and then the doc could claim that the nurse had put his family under stress and disrepute, and injured their beliefs and honour by not making sure her daughter respected said beliefs, right?