Not if they tell them the insurance is still in force; it's only the address that is changing. Why do you go out of your way to make up problems where none exist?
Okay, will this make everyone happy?
Dear Hospital Billing Office,
Effective xx/xx/2018 the address for Susie Blank will be 123 Maple Street, Anytown, State, 12345. The address for John and Mary Blank will continue to be 456 Oak Street.
Susie, Mary and John all continue to be covered under insurance Policy xxx123456789.
Really, this isn't as hard as you are all trying to make it.
OP, if your insurance is through your or your husband's employer, I strongly recommend that you notify their benefits office of Susie's change of address so that they don't inadvertently change it back to yours.
Two things I am tired of typing: 1.) A wrongful termination does not mean that you were fired for something you didn't do; it means that you were fired for a reason prohibited by law. 2.) The above answer, whatever it is, assumes that no legally binding contract or CBA expressly says otherwise. If it does, the terms of the contract apply.