At least here in America, shock jocks like that would have gotten in BIGTIME trouble because of the HIPA law federal and any state laws to the effect.
That would be the nurse’s problem, not the shock jocks. The ones holding the information are bound by confidentiality, not the ones fishing for it.
As for a lawsuit, what would she sue them for? It’s her responsibility to do her job correctly. All they did was give her the slightest bit of a nudge. Also, apparently her job wasn’t even in peril. Did they so much as scold her? What would be the damages? Emotional suffering, or sonesuch nonsense?