Here's the words to the song "Born To Hand Jive" from the Broadway musical and movie
Grease

Born To Hand Jive

Before I was born late one night
My papa said evrythings all right
The doctor let my ma lay down
With her stomach bouncin on the rack

cause a bebop stork was about to arrive
Mama gave birth to the hand jive

I could barely walk when I milked a cow
When I was three I pushed a plow
While chopping wood I moved my legs
And started dancing while I gathered eggs

The townfolk laughed
I was only five
Hell out dance em all
Hes a born hand jive

Born to hand jive baby!
Born to hand jive baby!

Now you can hand jive baby
Oh can you hand jive baby
Born to hand jive baby
Born to hand jive baby
Oh yeah! oh yeah! oh yeah!
Born to hand jive oh yeah!


http://www.icelebz.com/lyrics/olivia_newton-john/grease/born_to_hand_jive.html

There's no information about who wrote this song and composed the music.

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Here's a link to Born to Hand Jive from Grease (from the movie)

And here's a very short clip of Born To Handjive-Grease on Broadway

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Presumably, the "Willie and the Handjive" song was danced this way too.

The summary of the clip from the movie reads:
"Who hasn't never ever danced [this]once?"
-snip-

I'm raising my hand.