Sagar
This one is easy. Some college friends and I built a very effective can crusher in college to solve our large empty beer can problem. We simply used a pneumatic cylinder with a large washer attached to the plunger. After we supplied pressurized air from our air compressor and connected the cylinder to some solenoid valves, we had a crucsher. Also, at the base of the crusher cup we added a small pneumatic cylinder style clamping device with a lever arm that would flip the crushed cans into a recyle bin. It worked great and always added some enterainment to our parties. I can demonstrate it if you would like, it's stored somewhere up in my attic at home.
Stepper motor and solenoid are overkill and somewhat expensive. There are plenty of surplus DC gearmotors, you can use a scissor jack powered by a gearmotor to crush the cans. To automate the process use a relay for the motor control and a switch that reverses the motor when it is done crushing the can. You can probably add some sort of linear actuator to push one can at a time into the crusher to make it even more automated.
I know one thing that helps when crushing a can is too slightly crease the sides of the can before hand. Maybe make the cylinder that holds the cans have a bar smack the sides before crushing the can.
I'm not sure all of the goals, but to a degree this is re-inventing the wheel. Why not a simple "magazine" for the empties that drops them between two large rollers that will crush the can to your desired thickness depending on the gap? If you need to count, a simple switch that trips when each can passes through the magazine or count the number of cans crushed in each rotation? Dangerous? perhaps, but that is kind of how the recyclers will crush the cans for transport.
BTW, you can crush them smaller if you don't think of squashing them end to end where they are designed to resist crushing. Crush them from the sides and it will work much better.
A fast and fun way to SMASH those cans. Connect a 20oz CO2 bottle to a pneumatic piston that would be activated by a fun grip button. Load the cans via a magzine style loader. The smash cylinder could have a hole at the bottom back end to let the smashed cans fall freely away. The button (trigger), control board and valve could theoretically be harvested from a cheap electric paintball gun.
With my way you can crush multiple cans at once.
You have a metal bow with 1 open side and a hinged top, have a thick metal on the open side with a course threaded rod in the middle with a strong motor attached to the rod and when the motor is turned on the plate is forced into the box crushing how ever many cans you can fit in your box.
While I like Davids idea I'm thinking hydraulic may be a safer way to go. We can reuse some rain water for this idea if necessary. I like the solenoid as well and I want to expand on it a little more. Self admittedly I'm not an engineer or a physicist, but I think the solenoid just needs a more steady plus constant current and voltage applied to it. Put it in terms of the image supplied and you can turn the solenoid into a hydraulic ram. This is just a musing though and possibly over complicated. It would require electrical synchronization between the two rams via a transformer to up the voltage to the solenoids. AGAIN i'm not an engineer. I'm just going on what I've seen. Let me know what you think. 

Does anyone have an idea for an easy way to tear or cut off the ends of the cans? (so your left with a tube).
Make aero tubes out of the remaining tube (roll the end of the tube back over/under itself). Throw to use.
Remove the top (side with opening) and roll the other end over itself to remove the really sharp edge and use for glasses, or storage?
Roll threads into the can tube and screw many of these things together making longer cans?
Again cut out both ends, push the tube through a shape to reshape the tube into a shapes like cookie cutters, stars, clovers, hearts, maybe some really lightweight (toy) construction components?
(I thought maybe a center shape and external rollers).
Any way to cut out the sidewalls and make really weak really light springs (multiple spirals cut through the sidewalls).
Make cool reflectors (or pencil holders) by putting a large knurl pattern into the side of the can tube wall; anyone have an idea for an easy way to do it?
Is there an easy way use to roll the tube around itself as you collaspe the can?
I am thinking vaccum using a bar code scanner to know when cans are in place.
instead of crushing cans, the most compact size can be created with melting the metal.
use non flamable bowl,wire and hi frequency grenerator, place can in the middle of wire loop then true wire voltage metalic can quickly heats up and melts [done] this efect in physics known as induction.
see attached graphical picture: