We express available incident energy in calories/square-cm. I'm a bit
confused about the square-cm. I know how to convert watt-seconds to
joules to calories of heat. How does the square-cm factor into the
equation?

Think of Watt-seconds as energy released, when you talk about J or Cal/cm2 it is refering to that heat on a recieving surface, so to measure that you need to define the area of the recieving surface.