Thanks for all the input...I think I'll try a couple methods and see what works.
One thought I initially had for draw boring would be to insert wedges from both sides. Like one would in a wedged scarf. That way the peg hole would be filled on both sides. That would weaken the pin too much though if it was only 1" x 1". I would probably have to make it rectangular to avoid failure of those wedges. Or just not draw bore them and use one square piece.

Don, what do you mean by disturbing aspects of using the chisel from a mortise boring machine. Other than otherwise reputable timber framers whacking a chisel with their nail hammers eek! I agree that it probably isn't going to too hard even with a corner chisel. Just out of curiosity, what was your application for square pegs?