I'd never heard of this song before you two posted about it. However, I've played the clapping game since I was in kiddiegarter. I found the clapping game described online as follows:

Two friends can clap and chant together almost anywhere.
In the following rhyme, the clapping pattern is --

1. Clap own hands together
2. Cross arms in front of chest
3. Clap own hands together
4. Clap hands with partner three times

We did several different rhyme sets to this, but never the Mary Mack verses.

My favorite had the line: "My boyfriend gave me peaches, my boyfriend gave me pears, my boyfriend gave me fifty cents & kicked me down the stairs." (Only I always said that I kicked HIM down the stairs.)