On my Briggs and Stratton 5 horse engine the rewind "catches" while the engine is running and the starter rope shoots out and it squeaks.

Remove the blower housing, remove the starter ( in the center on top of the crankshaft), and either replace for about $15, or you may try to open it up, clean the inside, lubricate the ball bearings, use emery cloth on the top of the crankshaft, lube this area also, and reassemble.
Also, I did not see this question in the lawn and garden forum. It may have been there and I overlooked it.