You are probably going to use the 5 3/8" squares to make half square triangles.
If you want to sew two triangles together to make a square, you start by cutting squares cut that are 7/8" bigger than the finished size of square you want to end up with. Cut it on the diagonal, sew them together and you will get a 4 1/2" square.
4 1/2 + 7/8 = 5 3/8