"Napalm Sticks to Kids" is a call and response running cadence occasionally used in the U.S. military. The earliest recorded example is from 1970. A version containing a third verse is found in the final draft of the script for the movie An Officer and a Gentleman, but only two verses made it into the film. The line of "Dow Chemical don't give a shit" was changed to "Chemical Service don't...", most likely to avoid liability.
Recorded versions
tape obtained in 1970 at Phuc Vinh, South Viet Nam.
References
Napalm Sticks to Kids Wikipedia(Text) CC BY-SA