Does this help? Irish-born Joe Coburn succeeded John Heenan as Heavyweight Champion of America in 1862 because Heenan refused his challenge. See:

http://www.cyberboxingzone.com/boxing/coburn.htm

I found this by using "Coburn Heenan" as search arguments in Google. Is the other song a ballad about boxing too? We might be able to find something if we knew the names of the boxers.