how are you quilting? free motion or stitch in the ditch, etc? perhaps you are pulling on your quilt as you stitch. i can't think why it becomes uneven. i wouldn't square it up before quilting myself. usually squaring up isn't cutting off more than about 1/8 an inch. perhaps 1/4 in the border edge. not more.
there are many tutorials online showing how to machine quilt. and maybe your library has quilt books you can borrow. visuals help.