OK, officially decided on the 8x8 led matrix as a screen, easier and cheaper, although more shabby looking.
First of all . . . .
Why do you need to detect all 64 squares?
Last I checked, checkers only took place on half of them . . . .
You don't need to know the instantaneous state of all your switches, meaning you are able to use a method of polling the switches sequentially. Human movement is a slow thing.
Still not sure if the Arduino is enough if it has to run a screen as well, but maybe I'm not understanding the point of your project.
I was thinking for the matrix screen, since it is 8x8, the state of each reed switch is indicated so that: If the reed switch is closed, the led indicating that square lights up, if open, it doesn't. This way, the screen gives a live feed of the checkers location all the time.
My problem was how the reed switches would be wired. Would they all connect to its own input and output? Would it be one massive closed series circuit? What input and output type would be recommended for them? Online tutorials are handy when you have one reed switch, when you have 64, well...