This is a good question. I have a practice of only ever entertaining any tiny bit of discussion about colleagues in a positive manner
(the old adage, "if you can't say anything nice, don't say anything..."). "Oh, yes - s/he's very dedicated and spends a good deal of personal time helping others in the community" - is something I have conversationally said in response to a patient about a colleague, for example. If put on the spot (such as a patient making a "cheap shot" comment about a coworker), I do not entertain this and will say the nicest true thing I can say in response, or else pick a neutral reply like, "would you like to speak with someone about your concern?" If I get any inkling that the patient means harm or is being discriminatory in raising a conversation, I refuse to entertain that and I have plenty of ways to nip it in the bud, refuse to participate, and/or challenge them (and I do, as appropriate). As far as patient inquiries about a colleague's personal life, my reply is always the same - "I'm not sure, and generally-speaking I try not to talk about other people's personal lives, anyway." <sweet smile>
So, now that I have hopefully described my MO, I have to be honest: I can hardly bear for one more stinking thing in our environment to become toxically de-personalized and bound by restrictive policies that might as well make criminals out of us regular human beings as well as the ones with nefarious purposes. So I would hate it if it were some "healthcare crime" to mention a coworker. You know what I mean?
Racist and discriminatory people are the problem. People who entertain them are also the problem. People who gossip about others are the problem, too. But not everyone who would ever mention a coworker is part of the problem.
Perhaps it is wrong for me to mention co-workers in the complimentary responses that have been my habit in the past - - I will have to think about this. Never really thought about it before.
As to your problem - without labeling your concern as valid or invalid, it might still be worth considering how much trouble it's worth and what the chances of fixing the problem are. The alternative is that you could simply take these things as they come, win patients over by providing excellent care, let those go who prove they don't want to have faith in you by their actions, attitudes, and words, and maintain your own privacy as you see fit going forward. If there is a significant difficulty with the hospital's employees damaging your reputation or compromising your patient relationships through what amounts to slander, you could have a discussion with the CNO to that effect.
Take care ~