When you are tracing inside the square, you need to be careful that your pen or pencil is held at an angle so it marks close to the plastic. Just holding the pencil at a different angle for each side will get you off.
To be sure the fabric doesn't move while you are marking, it's a good idea to place the fabric on a piece of very fine sandpaper. That will hold it so the fabric doesn't "wiggle" as you trace.
Did you measure the square before you cut it to make sure that all 4 sides were equal?
Probaby the most important thing when you cut is to make sure that the angles are correct. If you have two rulers with markings, you can make sure the corners are square. Put one against the cutting line, and butt up the other one next to it. Any line on the butted-up ruler should match the line you traced. For example, if you place your cutting ruler on the line you plan to cut on the right side of the square, butt the other ruler up next to it and then check that any lines on the butted-up ruler match the bottom line you traced. (Sorry if that is as clear as mud.)
It's just that the secret to accurate squares is the angle of each corner. You can check that angle if you have two rulers.