Randall
About the only time you need bottle stopper blanks to be truly square is if you are doing inlay work - for example, celtic knots.
Here's the way that I do stoppers. It is absolutely general - there is no need for the blank to be square at all, and in fact I have made stoppers from random scraps of wood that were anything BUT square.
1. Identify the grain orientation. Ideally, stoppers should be spindle turnings with the grain running from top to bottom.
2. Mark an approximate center on opposing faces of what will be the top and bottom. This can be done by eye - not need for great precision here.
3. Mount the blank between centers, and turn to round. Turn a tenon on the end that will be the top.
4. Remount in a scroll chuck and face off the bottom. I prefer to make the bottom slightly concave so that the join between the turning and the stopper base will be hidden by the edge of the wood. An extreme version is this is to use a forstner bit to cut an actual mortise in the bottom about 1/16" deep, but I've come to understand that is really too much work - cutting a slight concave with a gouge is good enough.
5. Mount a jacobs chuck in the tailstock and drill a hole in the blank. if the stopper has a 3/8" mounting stud, a 5/16" hole is fine. Thread the hole with an appropriate tap, and then reinforce the threads by soaking them in thin CA. After the glue has cured, run the tap in again to clean up the threads.
6. Sand the bottom end of the blank, and apply finish. The purpose of this finish is only to prevent moisture from penetrating the wood as the stopper is used, and I usually just use a little thin CA for that purpose.
7. Remount the blank on a stopper mandrel, and turn to the desired shape. Sand and finish using your preferred finish. I normally use either WOP, a shop-made Danish oil (equal quantities of tung oil, spar varnish and turpentine), or pure tung oil.
8. When the finish has cured, attach the finished turning to the stopper base. Screw-on is simple. For the dowel and silicone sleeve type stoppers, enlarge the hole in the turning to match the diameter of the dowel, and then glue the dowel in the hole.