If it has the latest version of Flash but it's running Windows XP, perhaps that's the issue.
What I've found on this so far is more for multi-core CPU's running Windows XP and Flash, and these are more for video issues and Youtube related complaints than Facebook games.
One solution:
"Launch browser (IE for Firefox
Go to Task Manager
Click on the browser process.
Right click and Set Affinity
Un-Check every CPU except for the last one."
But that's for multi core. So let me see what else I can find.
-----
It looks like on some XP machines McAfee AV detects flash games as malicious attacks or intrusion attempts, and it freezes things (Whether it's just the browser or Windows entirely) when it tries to intervene, often before it can even get a message out.
To verifiy if it's a hard drive issue I'd recommend grabbing SeaTools or another free hard drive tester. Don't run anything that it says may damage your hard drive - you should be able to get a good idea from running the basic tests. But if it's the hard drive it wouldn't freeze up due to a single activity.
-----
Have you already tried completely uninstalling Flash and the subsequent plugins and reinstalling fresh to clear out possible upgrade leftover files? And have you checked your events logs to verify that there's not any errors occurring that Windows may be keeping track of for you?
One person had a similar issue on a Dell machine and rebooted to VGA mode through the F8 startup menu - They say that it fixed everything, proving that it was a video issue. Unfortunately, if it's onboard video, once it starts to go, it may be a sign of things to come as far as your motherboard is concerned. So while a PCI video card would fix things temporarily, if you see bulging capacitors when you pop the case, it's time for a replacement.
But the most common solution seems to be completely uninstalling and reinstalling Flash and its constituents.
http://get.adobe.com/shockwave/otherversions/