Yes, and the law states that if its private property, they can prohibit anything they want, same as anyone else.
What law? Where does it state this?
So if it's private property and the sign was there when OP walked in, then the hospital has every right to ban him from the property.
Yep. They can ask anyone to leave or even not to return with or without reason.
But they have no power to prohibit objects like you claim.