I would never rip fabric to square it. It damages the fibers up to 2" from the rip. This damage is not necessarily visible to the eye, but can be seen if the edges are examined under a microscope. You do not want one of these edges in a seamline in your quilt. (Note: It's not as bad to rip along the lengthwise grain. I will rip in that direction for a border, but even then I plan on eliminating the first inch or so near the rip.)
For garment sewing it is important to cut exactly on grain so the fabric will drape nicely over the hips and, er..., other body parts. It is not that important to be exactly on grain when cutting pieces for a quilt. You just want to be close to on-grain so that the edges of your pieces don't distort from handling. Bias edges are the most likely to stretch and distort from handling; the closer you get to on-grain, the less instability you have for the fabric edges.
For a swap, all you need to do is to cut reasonably on-grain. It's not really helpful to pull on the bias to straighten the fabric before cutting; there will be a tendency for the fabric to go back to its original form. I don't do swaps, but when I cut my own fabric I usually just line up the selvedge edges to be roughly parallel. I then use a ruler to help me make the first cut exactly 90 degrees parallel to the fold. If this requires cutting off up to an inch of fabric or so in order to get a straight line, that's fine with me. If it's a small piece of fabric and I will lose 2" by doing this, I usually "fudge" a bit and let the selvedges not line up perfectly in order to save some of the fabric. The little bit off-grain that these cuts will be do not affect my pieces.