I dont know about mint, but debian does security updates automatically and just politely asks for a reboot now and then, perfect for an "unattended" device.
Yeah, Mint can do this too. I've set it up on a few of our computers, but it was a while ago so I can't remember exactly how. It was very easy though. The only downside was Firefox insisting on a restart after an update, but apparently that's been fixed now :)