Tell them that the second they reimburse you for the server they can not only get a login, but they can become responsible for its maintenance and security and they had better be sure it has a solid uptime. That only seems reasonable.
Nope, I'd just quietly get the MAC and blacklist it. No network traffic for you. If I'm asked to buy a CalDAV server, I'll buy real server hardware and run it in a real server room. Not under some guy's desk where the custodial staff can kick the cable.
If you're feeling REALLY confident about your value to the hospital, feel free to bet on your clout. But if that's the case, Management probably would have paid for the server if you asked.