I cut my rough stock on the table saw at 3/8" square. But my input hole on the die was 3/8" . Keep in mind that from corner across on a 45° to other corner will be 3/8 (.375) x 1.4142 = 0.530 (17/32). But it had to fit into a .375 input hole. That meant I needed to shave off .078" off each corner with my hand plane before chucking up into my drill and running into the 3/8" input. I did this in a matter of minutes.
Next time I will cut 5/16 (.312) rough stock. Across its diagonal it will only be .441. This minus .375 means I would only have to shave off .031 (1/32) from the corners. Basically a pass or two on each corner with my hand plane. 
Yes you can use square blanks from your bandsaw. You can cut them to 5/16" or 3/8" square. The corners will have to be knocked off to fit into your round input hole Either way will work. 
Hope this helps. 
Cheers.
Rick