Am I using the crank extractor correctly?
I don't know about this this particular crank extractor, but the one I have came with two different end caps, one for ISIS/Octalink and one for square taper. Check that the end cap you are using fits inside the square hole in the crank, otherwise you will just strip the threads.

After you are sure you have the right tool, use more torque. If you have a piece of metal pipe at hand, use it for extra leverage, or get a longer wrench. Turning clockwise is correct.