You need the different cast on sts because of the different st patterns used so the squares come out the same width. Garter st stretches widthwise, so you need less sts for it. Moss st is a little more like ribbing which pulls in, so there's more sts for that one.
To make them in strips, going from square A to B is okay, you can just inc a stitch, then you need to inc 4 sts for C, and after that dec 5 sts for another A. I think you can do this by putting a garter stitch ridge between each of them and doing the incs/decs on that row. So you'd work your rows for each of them, but instead of doing the last WS row in pattern, just knit the row with the incs or decs. I think that would work out and make for a lot less sewing.