First, I'd just like to say thank you
for the responses!
After speaking on the phone, I found out the problem was a corrupt filesystem. Apparently, it required an fsck, and was prompting for the root password, which... doesn't exist.
Booted to the LiveCD, and made a few attempts at running the fsck from there. It corrected it enough that I could mount the partition and make a backup of the home folder, but still wouldn't boot, kept prompting for that non-existent root password and gave no alternative other than rebooting. Some quick Googling shows me that this has been an issue with Ubuntu for years...
Eventually, I gave up, reinstalled, and restored the home folder. I did get openssh installed, and the remote desktop feature (VNC server) works great, as well, so long as someone is logged in. (The wireless connectivity also depends on the logged in user, so my remote admin sessions kept getting interrupted when I rebooted.)
I asked specifically what he had read that was expiring, and he couldn't remember, said he'd wished he'd written it down. All he could remember was a message that something was expiring or terminating at midnight of April 30th... There's no telling what generated that message though, and I suppose it's a moot point, by now. Certainly the real issue was that the machine wouldn't boot due to the "fsck needed" issue, not because something "expired".
Anyhow, thanks again for the replies!