Sounds to me too like he wants a reciprocating coil gun in which the "projectile" is a piston. I suspect it is just to make a gadget out of curiosity and not for any use.
So, I would suggest he read about coil guns and solenoids. With a simple design, it seems one could alternate solenoids and get rotation for a crankshaft. One could even get rid of the crank and put the pistons on a cam (e.g., dynacam design).
Capturing the back emf to run the other piston seems unnecessarily complex and superfluous.
John