Anything is possible but this would require a huge amount of labor and money if you want it to work like the factory push button.
Push button starts work with a series of antennas/sensors around the car to detect whether the key fob is outside of the vehicle or inside. The car only starts if the fob is inside the car. If you were to retrofit something it would take much more effort than what it is worth to put those antennas/sensors in. If you didn't, the car could easily be stolen as someone could just press the button, start the car and go.
Push button starts aren't really all they're cracked up to be. I am currently running a Nissan Altima rental that has push button and I dislike it. The other day I was just trying to get it in to accessory mode without actually turning the car on and it was a huge task to try to do something so simple whereas all I would have had to do in my Cruze was turn the key halfway. Keyed ignitions are fool-proof and an ideal thing to have in a car. Factory push button starts have their own set of problems. I can't imagine what kind of headaches a retrofit push button system would cause.
t;ldr It's possible to do it but not worth the time or money just to be able to push a button to start the car.