If you let your cursor hover over the box it says, "You are subscribed to this thread." Now what that means, I have no clue.
Edit: In looking over the ones I have that are checked, it appears that once you have made a comment in a thread, you are automatically subscribed to that thread--otherwise no blue box. I still don't know what that means.
2013 Escape 15A (April), "Egbert"
Ford 150 XLT 5.8L