First you put your two knees close up tight You swing em to the left and you swing 'em to the right Step around the floor kind of nice and light And you twist around and twist around with all of your might
Spread your loving arms way out in space Do the eagle rock with style and grace Put your two arms out and swing 'em back And that's what I call Ballin' the Jack
I would hazard a guess that this is a tin pan alley song composed by someone to incorporate a well known slang expression.
I always assumed that ballin' the jack was the same as a going out on that old tenderloin but I could be wrong.