This would be the hospital with the giant signs at each driveway, proclaiming that their campus is smoke-free as a means of promoting better health

On one hand, it is private property, and they can do what they please. On the other hand, it contains a public bus stop, and said buses have bike racks on them....ergo.... I have appointments and meetings at that hospital, and it irritates me that they would deny me the right to ride my bike there should I ever choose to. Though it may explain why I've never found a bike rack there.

I do have to wonder if in fact it's a hospital policy, or if the guard was mistaken. I think I'd ask to see it in writing Like the others have said, a bit of well-placed publicity should clear up this nonsense.