For a start you need to measure the current the motor takes, under a variety of conditions - including stall conditions with the battery pack you intend to use. The stall condition current will probably be limited as much by the battery pack as the motor, and will certainly be well into double figures.
A 600mA IC is way too puny, you need to be looking at fairly hefty FET's, the reversing in cordless drills is usually done by a DPDT switch, rather than an H bridge - which would cost a lot more to do, and lose extra power in the extra FET's.