I can confirm that the cause is a combination of the Google Chrome browser and switching the media upload to Browser Uploader instead of the new Flash uploader.
I was getting HTTP errors all of a sudden using the Flash uploader so I switched it to the browser uploader. The white box immediately appeared but I didn’t put two and two together.
When I started troubleshooting it and disabling some plugins didn’t fix the issue, I decided on a whim to see if it was the media uploader. Once I switched it back to the Flash uploader and restarted Chrome, the little white box disappeared. Weird thing. Luckily the HTTP errors have gone away too.