How much room does a fridge coil need to dissipate heat?
I can't say for sure whether your installation will work, but heat dissipation could be a problem. The fridge innards you plan to use were meant for a refrigerator that has free air exchange with the rest of the house. Sealing it up under a counter with no free air movement will hinder its performance. Refrigerators that are built for under-counter installation have the coils mounted on the bottom (with a grate in front of the coils) for this reason. You could try mounting the coils on the side or bottom of the cold box, closer to the open air of the kitchen, with a vent panel as Steven suggests.