This is a pic of one of the bolts. But I don't see a channel for it to turn/slip out of. The toilet I bought needs longer bolts.
If you have access from below it really is a quick job to put a new flange on. But honestly if the one you have is that solid and has worked well (doesn't clog a lot), then I would simply just throw on a bolt extender.
Note: Your bolt looks like it extends pretty far so not sure what toilet needs more than that but gave you an answer.
You could also try to snap the bolt off, or if it will unscrew, if it is rusted or corroded, and use a lag type anchor with standard threads on top. However, for the wax ring to seal properly, Id say best bet is a new flange. You can get retrofit flanges that bolt down over what you have, basically creating a new flange like if yours broke, or the floor was raised by tile etc.
Like any of these items: http://www.homedepot.com/p/DANCO-HydroSeat-Toilet-Flange-Repair-10672X/204403879