When I have to sew a square in a square, I use the same theory as making half square triangles by drawing a line diagonally across the square and sewing 1/4" each side, then cutting them apart.
Instead, draw the line corner to corner on the front of the corner square before you've cut them apart to make triangles. Line the center square up to that line and sew the 1/4" seam. Turn it 180degrees, and sew another center square on the other side of the line . THEN cut them apart along the pencil line. By doing this you've sewn down the bias edges before they are cut, so there is no problem with stretching. Here is a tutorial that shows this technique.
http://tjzoriginalz.blogspot.com/201...ways-told.html