Turns out I was using the wrong event handler... Adding the following line to the quiz_do_top did pretty much everything I was hoping for.
Code:
<body onbeforeunload="document.forms['quiz_form'].submit(true)">
It wasn't perfect... you could still click Back in IE7 and it wouldn't submit, but it prevented circumvention via closing the window in both IE and Firefox and by the back button in Firefox. (To fix the oversight, I used a pop-up window with the toolbar disabled to open the quiz page and added a Javascript function to disable right-clicking.)