I think there are probably a couple of things going on here (in Chapter 20).
First, I think that this plan satirizes the feud that was going on between the Grangerfords and the Shepherdsons a couple chapters before this one. The two families had their very own Romeo and Juliet story going and I think this refers back to that.
Second, I think that it is meant to show how stupid the Duke and King think country people are. They think the audiences will buy the idea of the "king" being Juliet.