Maybe you mean something different than what I'm used to by 'strain gage', but the ones I've used would not work the way you're describing. You would need to mount it to a surface oriented parallel to the direction of thrust. Perhaps you're talking about a load cell instead?
For cheaply monitoring the torque on the spindle, how about measuring the electrical power going into the motor? It wouldn't account for a variety of losses in the motor and drive train, but it would be simple. You could measure the idle current and then the current while drilling. Or what about applying a strain gage to the shaft of the drill to measure torsion of the shaft. You could lead the wires out via some kind of ring and spring loaded contact setup.