I've been seeing what sounds like a similar problem (Windows application "BEX" error when attempting to close Flash Builder via its [X] icon).
(Win 7 Enterprise, SP1, 64-bit, i5 + 8GB RAM), Flash Builder 4.6, Java 1.7.0_07)
I tried the following:
-- Launching Tour de Flex (from the task bar icon), and then updating AIR to the latest version
-- Right-clicking on the Flash Builder task bar icon and then selecting "Run as Administrator"
-- Right-clicking on the Flash Builder task bar icon and then selecting "Troubleshoot compatability", then applying the compatability fixes it suggested.
Somewhere along the way, the problem went away. I then ran the compatability troubleshooter again and (I think) removed the compatability changes (since windows security was asking me whether I want to let the application update my computer every time I launched FlashBuilder--annoying!).
The problem still appears to be fixed--but only with a whole ten minutes testing. I'll just have to see if the above is really a fix/work-around.