Damn, the machine rebooted again last night with 2.4.17. Half an hour later, it hung for 2.5 hours until I could power cycle it.
I brought it back up with 2.4.16 and grsecurity. It ran fine on that for a while, so it'll be interesting to see whether this stops the spontaneous reboots and hangs.
The machine is logging nothing strange at all prior to the reboot. There seems to be no weird network probing activity going on, the machine isn't under load. It's very strange.