I sanded then painted my kitchen cabinets with a primer by Behr and two coats of a gray eggshell finish latex paint by CIL. Now I notice that the finish is easily scratched. Will this paint harden up over time, or do I need to put some kind of clear coat over it to make it durable in the kitchen?
All paint finishes will harden over time, so yes. However, thicker coats of paint take longer to cure.
You used latex paint - this is a softer film surface than alkyd paints. For this reason, I prefer alkyd for kitchen cabinet doors, although latex paint can be applied successfully if it is built up in thin layers.
Rather than applying another coat of something on top of the paint, I suggest leaving the cabinets to cure for a time. Once the paint has cured then you can determine if the finish is hard enough (durable enough) for your environment.