With the external monitor plugged in the screen goes blank and the machine no longer responds to keyboard or mouse input.
Do you have another machine to test if the machine is actually frozen (using ssh to open a remote connection)? That means the difference between an X issue and a kernel issue.
I have to power it off and back on. I wonder if the 1280x768 laptop screen resolution is a problem?
Only if X isn't configured properly
When plugged into an external monitor, it works about 50% of the time. I tried running it with and without the --quiet option. That does not seem to be the solution. Any ideas how to fix this or how to troubleshoot further? There are some generic errors in the X log, but they are the same when the game starts and when it doesn't.
Have you tried the magic Sys Rq reboot sequence hold alt+sysrq (Usually found on the print screen key) then with those keys pressed, press s then u and then b which will sync your filesystems, remount them read only and then reboot.
If that doesn't work ssh into the machine and
echo s > /proc/sysrq-trigger
echo u > /proc/sysrq-trigger
echo b > /proc/sysrq-trigger
If that doesn't work you have a kernel issue, if it does you have an X configuration issue.