I can only speak for the ones I've seen here in the South. I've never seen them do the little shuffle dance. They just stand up, crouch down, stand up, crouch down, etc. all while shrieking and chattering at each other.
Then one will suddenly launch itself at the other, they'll tangle for a few seconds and then one will break away and shoot up the nearest tree with the other in hot pursuit.
I see squirrel fights in my front yard and my neighbors' yards all the time. Mostly when they've already taken it to the treetops, leaping from branch to branch and tree to tree while chittering loudly at each other.