I can only remember one pair of lines, and it's bugging the heck out of me.

Running cadence:
"The Navy swims and the Air Force flies,
But the Army will run you til the day you die!"

Can't remember anything else. Anyone got anything? So far my google-fu is weak.

Just a guess, but it sounds like a verse to the traditional "Sound Off" cadence. If so, I think the rest would follow something like this:

Sound Off
One, Two
Sound Off
Three, Four
Sound Off
One, Two, Three, Four
One, Two (pause) Three, Four

There are/were infinite verses to the Sound Off cadence. Often they were just made up on the fly by whoever was leading the group - either an officer or NCO.

What the hell are we doing here?
Lets go out and get some beer.
Icy cold with lots of foam.
Like the ones we had at home.

one
two..etc

ain't no use in looking down / ain't no money on the ground I don't know but I been told/ blah blah blah is made of gold