The reel is a Progress. The good news is that the problem was actually with the spool, not the screw. I have two of these reels, one with the clicker, one without (the one without the clicker doesn't have a screw to hold the spool on ). I loaded the line on the one with the clicker.
So, I took the empty spool and put it on the reel with the clicker and tightened down the screw. After about what seemed like a thousand turns the screw was still just as tight. Upon further side-by-side inspection it appears that the spool I loaded has a bit of a raised area that contacts the screw causing it to come loose. Now my project is to move the backing/line/leader over to the other spool and go fishing.