Okay, it's been a long few days for me so I apologize if this comes across as snarky, I don't mean for it to...but ya'll are making this waaaaayyyyyyyyyyyyy more complicated than it needs to be.
I do not work in a hospital. But I have an ex who not only participated in ministerial visits (aka clergy care) in numerous hospitals over the years, he was the director of the program in two different states, president of not one but several ecumenical societies that participated and funded these types of programs. I saw and participated.
The question at the center of this debate, while not a 'requirement' of a clergy care visit IS an expected question. Protestant clergy care began in most hospitals as the opposite answer to Catholic last rights and many people DO find solace in a clergy visit, the ex even had some athiests that while they didn't believe in God per se, they were certainly glad to have somebody to talk to. Being in the hospital is scary enough, surgery or a terminal illness is even scarier. Nobody disputes that having a minister visit is ultimately a good thing, I'm sure...what at dispute is this question...which almost every minister I've ever met who dealt with clergy care asks. Just about every minister in clergy care DOES ask this question, no matter how it's worded, and it's basic logic to them, it's an expected question within their realm, to make sure the patient has it all taken care of 'before its too late'.
Now, to the many open minds of the world, this is probably offensive, they may even think said ministers are trying to proselitzye(sp), which, in essence, they are, and I've heard it come from the mouths of many different ministers - Baptist, Methodist, Presbyterian, Mormon, you name the denomination or group, they've said it in a clergy visit. Why? They're trying to make sure that soul is destined for Heaven and it's a requirement of their job. If somebody thinks that's offensive - don't check the box when you admit. (I mean "you" as in the general public, not you, the OP)
Almost every hospital in this country has a religious preference box on their admission paperwork. Make sure that a note is placed there that you (general public) want no ministerial visits. And if you get one, there's no point in being offended by such a question because they're not singling you out, they say it to everybody, just ignore them and say I don't discuss my religious preferences with people I don't really know and move on to something else. This doesn't have to be as complicated and drawn out as it's developing here. (and if a person does feel like they're being singled out by a clergy care visit then per chance they just might want to take a closer look at what they consider to be their spiritual walk, if they have one, conviction comes in interesting responses.)