Does the drive have metal threads encased in the plastic of the HDD itself? If so, Id be willing to bet the thread had dislodged and is just spinning.
I'll give it a shot.another thought. If the threads are stripped and the hex head or what ever it is there could ya use and grab with small needle nose pliers or 6" side cutters and pull it out?
What wolfe first mentioned should work if you do it right. If you can get something under the screw head and use it to lever enough force as you turn the screw it should come out.
Last ditch option would be to take a drill or dremel to the screw head and completely remove the head of the screw.
If it is spinning, try turning the case around so that gravity helps it out of there.
But if it is still threaded and you just can't turn it, find some old worthless screwdrivers or something similar, put a little dab of super glue at the end, and let it dry and then try turning it. Actually worked for me a couple times. Only really works on small screws.