If it is really cross stitch (as opposed to needlepoint) then you should have no problem squaring it. Cross stitch is worked in crosses (duh!) on an evenweave ground fabric, of which a lot shows in the design area. This differs from needlepoint, which is worked in a diagonal stitch covering an openweave canvas, none of which shows in the design area.
Cut a dummy mat (in case you have to cut the real one a tiny bit larger/smaller). Cut the backing board the correct size. We use rag mat ATGd over a/f corrugated. Put the work under the dummy mat and on top of the backboard in its approximate correct position. Lift up the dummy mat and peek to see that there is enough fabric (about an inch at least) to lace in place. If not, sew similar fabric to work edges. Cut a 'donut hole' in the backboard of a size that a) will accomodate the work and b) will be covered by the mat. Trim 1/8" off two sides of the fallout so that the fabric will have a place to fit when it is wrapped around the 'hole'. Lightly sand the edges of the donut hole to remove sharp edges, to protect the fabric from abrasion.
Put the hole back in the donut, lay the fabric in place, using the dummy mat to figure placement. Remove the dummy and the donut, holding the fabric in place on the donut hole with your hand. Clip on each side with a bulldog clip or a binder clip. Lace fabric around donut hole in one direction. Remove clips, test fit in donut, squaring fabric by pulling it around donut hole as necessary. Lace in other direction. Test fit again and make adjustments as necessary.
When it is square and located properly, turn the donut and hole combo over and use a few pins to hold the hole in the donut. Cut the real mat, making any necessary adjustments as to size.
Hope this is clear! I am more of a 'show' than a 'tell'...