Yes, you would read in a serial bit stream from the shift registers that would represent the state of all the squares into a signle input pin on the microprocessor and then store them into an array in your software program.
That sensor you show might not be any good at discriminating between colors just the presence of a stone or not and the size doesn't look to compatable with the large number of squares you have to sense. I'm not sure what would work on a budget basis.
Lefty