I only cut binding on the bias if I have curved edges (like rounded corners, a round quilt or scallops) or if it is a show quilt. In my experience, the challenges you face (when mitering corners and joining the edges) won't be any different if you use binding cut WOF (width of fabric, aka Straight of grain binding) or on the bias.
Bias binding allows you to sew curves and the tight ins and outs of scallop. It also provides better wear because the edge of the quilt is dealing with a criss cross of the warp and weft threads of woven fabric and WOF binding the wear is only on a few threads.
If by square binding you mean binding cut WOF, what makes you think it went out of style? By far many more quilters bind with WOF strips because it uses less fabric and is much quicker and easier to make than bias binding. I like working with bias binding but I don't always want to take the time to make it and I don't always have the extra fabric to make it. It does seem to "hug" the quilt better as I flip it to the back to hand stitch it but that little nicety is not enough for me to do bias exclusively and I think most quilters feel that way and will only use bias binding when dealing with curved edges.