I know I am late on this but did you check to see if your jigsaw blade is exactly perpendicular to the footplate? being off even half a degree will cause the cut to not be square. Also just out of curiosity how are you following the line? I assume that you are just keeping your blade on the line and not something on the foot plate. if your blade is slightly twisted in the receiver so that it does not sit facing directly forward. Your saw is going to naturally going to want to deflect from the line. You seeing the deflection will have a tendency to put side pressure on the jigsaw which will cause the blade to deflect. What you should do is twist the whole saw so that the blade is the parallel to the line even if the saw is not.
I would do a test on a scrap piece of wood. anchor a straight edge using clamps onto the piece of wood on one side of the cut. cut for about 12 inches keeping the foot plate butted up against the straight edge. if the jig saw starts pulling away you will know that the blade is twisted.
D