Squares are square, right? Yeah, I thought so too ;) until I had the chance to use different brands of cutters ;) Most of the cutters available on the market have slightly rounded corners. To make a checkerboard I suggest using the ones with square corners ;) I used Fine Cut cutters.
Before cutting the squares out, make sure that the paste doesn't cling to the board. If it does - flip it to the other side and allow the underside to dry for a few minutes.
After that time flip the paste back to the other side. Place the cutter on the surface of the paste. Press the cutter firmly all the way down to the board. Once the edges of the cutter reach the board make a few circular moves on the board (as if you'd like to rub the edges against the board). Lift the cutter. If the paste stayed in the cutter rub the edges with the surface of your hand (clean cut). Gently push the paste from the cutter. When holding the cutter don't press it from the sides - if you do, you will damage the shape of the cutter (it will no longer be a perfect square and you'll have difficulties with making an even checkerboard pattern).