The engines have to be destroyed; most common technique is to fill the crankcase with a goo that hardens into a ceramic, and run the engine until it seizes. That renders the engine block and major components unusable; it's theoretically possible to undo the damage to some components, but it would never be cost-effective for an old engine.

The *rest* of the vehicle can still be solid for scrap (windows, suspension bits, drivetrain, etc). I believe the origonal 'cash for clunkers' bill called for the transmission to be destroyed too (probably by replacing the transmission fluid with the same goo) but that was deleted due to pressure from used-car-parts dealers, on the grounds that those are the only parts of an old car worth much.

Technically, though, it makes a bit of sense: if you wreck so much of the vehicle that they loose money disposing of the remains, junk dealers won't take them. You'll either have to pay the dealers to dispose of the remains, or worry about them dumping the hulks in the nearest lake (or worse, both).