Greetings, all! I hope everyone is well! I was wondering about the Down By The River handclap circle game ever since I taught my youngest sister it. So, I’d like to contribute my location’s variation on the rhyme! I learned it from the older girls at school (in RI, USA) sometime around the later 2010s, and I believe it went something like this:
Down by the river with the hanky-panky
Where the bullfrogs jump from bank to bank-y
Saying ee bop ee bop bop, skittle diddle lollypop

I pledge allegiance to the flag, Michael Jackson makes me gag/is a fag
Coca Cola has caffeine
Now we're talking jelly beans
Jelly beans are out of style
We've been talking for a while

Ee bop ee bop bop, skittle diddle lollypop
One, two, three, four, five, six, seven, eight, nine, ten!

I’m not exactly sure how the middle part (paragraph) goes, as it’s been a while. I feel like it was longer, too, and that sometimes we sang “Billy Jean” instead of “jelly bean,” but I am probably wrong.
I hope that one day, we can collect all of the variations and keep them going! Some of these are cool and others hilarious! <3