Parallel Anthology Track 14
Track 14: Kelly Harrell and the Virginia String Band - My Name is John Johanna Victor 21520-A; Recorded in Camden, New Jersey, 23.3.1927
The Roud Folksong Index classifies this song as a version of Roud 257. A Roud number search for this song  returns a number of records of versions collected prior to Harrell's recording, dating as early as 1906, and across a wide geographical area (Kentucky, N Carolina, Missouri, Virginia, W Virginia).
According to the Where Dead Voices Gather blog, "'My Name Is John Johanna' aka 'The State of Arkansas' is a humorous song which comes from the minstrel stage, likely dating the song between the 1820s and the 1840s, during which blackface minstrelsy was at its height. The song tells the story of an unfortunate young man who finds himself appalled at the living and working conditions in Arkansas. The state of Arkansas was admitted to the Union in 1836. This song draws on fairly typical frontier stereotypes, Arkansas being largely untamed wilderness during the mid-19th century."
It seems likely that this composition is public domain, and there is no firm evidence to contradict this assumption as yet.
In any case, according to wikipedia, Kelly Harrell died in 1942, so even if this recording is Harrell's own copyrighted version of a traditional folk song, this composition has been public domain since 1st January 2013.