I wonder if you could hire a rope guy (not sure what the correct term would be). You know, the guy that can take a lasso and spin it around, step through it, etc. That seems like it would be good fun to watch. You could also couple this with the clowns so that they could be roped.
Yes - the key to a good square dance is a great square dance caller. Our kids did have Square Dance as a unit in PE before the event AND the gym teacher came to the event which helped facilitate the dancing. The kids enjoyed teaching their parents how to do it. The caller we got also mixed things up with standard line dancing, like the electric slide which many people already know how to do. We also have a local square dance club that sent a "square" of folks to demo the dances!
Something that might fit the theme is donkey basketball (for the adults). Very fun to watch!!!
We had this at my company and there were multiple teams. This is more popular than you might think and there is likely to be a place close by that can bring this in (not sure the type of fee).
We had a squaredance caller last year and it was a flop. The kids had no clue and weren't interested. We had asked the gym teacher beforehand if she could start showing the kids some steps - she agreed and then got sick and was out for two weeks.
One other suggestion is ask the caller if there's a local club that would be willing to come and dance and pull the kids into it slowly.
How about change the theme slightly and call it a Hoe Down? Square Dancing is something everyone can do! Get yourself a good caller, find a local square dance group to come give a demo an get folks out dancing! You could still have many of these other activities along with the square dancing!
You could also do pig races. In Texas where I live, pig races at the rodeo are a huge crowd favorite. Obviously, you couldn't have actual pigs...but for my son's rodeo birthday party we did at home, we used pink balloons and fly swatters. We blew up the pink balloons and taped on a pink construction paper nose (with 2 black dots for the snout). Parents each got a balloon pig and a fly swatter and they had to "swat" at their balloon pig to make it move. The first pig over the finish line won - we raced 4 at a time - but you could do whatever you wanted. We posted the fastest times on a giant piece of paper. The fastest pig and parent won a prize. This was a hoot!!