I'm also getting both as per my hospital's policy but I find your DH's facility's statment really odd. I thought it was common knowledge that there are no guarantees even if you are vaccinated. What I was taught to tell my patients is that if they do get the flu the thinking is that it won't be as severe as it might have been.

It probably is a violation of rights except as karousel wrote I'd prefer to know that my compromised/hospitalized loved one wasn't being cared for by a carrier of one of these diseases. As health care professionals kind of know there are certain risks involved in our jobs I guess.