Don't be mislead by incompetent journalists
Facebook has absolutely nothing to do with this. As usual, the journalists (not just ElReg's) reporting a technical issue have screwed up.
My guess is that the only one of the people mentioned who is playing Mafia Wars on Facebook is the "anonymous defence official" - the source quoted by Associated Press. And since "military installation hit by Mafia Wars virus" sounds sexy, all the stupid journos have jumped on the bandwagon.
In reality, the computers of the drone program have been hit by a keylogger. A keylogger logs whatever the user types - usually as login passwords to web sites. Yes, it could be the password to your Mafia Wars account. Or to your GMail account. Or your bank account (which is usually the real target). Or whatever.
It does not mean that you are playing Mafia Wars on that computer. Or that you're using it to check your GMail account. Or to do Internet banking. The only thing it means is that your computer has been infected by a keylogger.
Which is, actually, much worse. If one of the infected computers is used to login to a classified account without using anything besides a user name and a password (e.g., smart card, a biometric scanner or whatever), the attackers now have access to that classified account.
How did the military get infected? Certainly not by playing Mafia Wars. Most likely, the infection came from an USB drive. The drone pilots often bring updates to maps, etc. on USB drives.
Why wasn't autorun disabled on these computers? Incompetence.