I agree with the others for cutting 3" strips and then cutting the squares from that. When you cut, are you positioned above the ruler? I noticed that when I sit too low, I don't get enough pressure on the ruler and it tends to slip. It may also help to cut single layers of fabric rather than multiples.
When you say the squares are off by "a hair" is it enough to distort the row, or can the next row pick up the slack? You have a little wiggle room.