So, where did this song come from?

The song's origins come from a minstrel song titled "Billy Patterson," originally written by Dan Emmett, of blackface minstrelsy fame. That song originated from a popular American joke dating to early in the 19th century revolving around the phrase "who struck Billy Patterson?" Though the origins of the joke are unclear, the earliest known reference to the joke was in 1837, in a story detailing the Broad Street Riot. The article, which is clearly mocking the Irish for the riot and appears to be using "Billy Patterson" as a joke already, detailed the attack on Billy by an unknown assailant and the dialectically-charged lamenting of his friend, Patrick McManhonie (in later texts his last name is sometimes written as just Manhonie). As the 1830s turned to the 1840s, the name's status as a joke grew. Frequently, variations of either "Billy Patterson" or "William Patterson, Esq." would be listed in the "Arrivals" or "Departures" sections of local papers, or in some other section of the local news. Eventually, the question became a sort of quip to represent something unknown, and was frequently listed in joking indexes of unanswerable questions or joke lists, like the following sarcastic list of items intended to draw visitors to a new museum: "One of the rocks which the man had in his pocket when he was in town... A bag of wind... [a picture] of the individual who struck Billy Patterson, A fine likeness of John Smith, [and] A daguerreotype of What's-his-name." ([1], [2]).

In 1859, Dan Emmett capitalized on the success of the joke and the popularity of his blackface routine to make a new comic minstrel song, Billy Patterson. In the song, Billy Patterson changed from his previous Scottish extraction to African. The song went as follows:

Dar was an old nigg dat got hit wid a brick
 Chorus.--Oh! Billy Patterson
He wasn't knocked down kaze his head too thick
 Chorus.--Don't y'e tell me, don't y'e tell me
De first word he said when he was come too
 Chorus.--Oh! Billy Patterson
"O, don't hit agin for dat will doo!"
 Chorus.--Don't y'e tell me, don't y'e tell me

 CHORUS
  Bill Patterson rode bye, Old Bill your horse will die!
 "He dies, I'll tan his skin; He lives I'll ride agin!"
  I'll gib ten dollars down, an leab dem in my will
 If any one can show de man dat eber struck old Bill.

I eat up de goose dat rais'd da quill,
Dat wrote de question: "Who struck Bill?"
I work'd at de kiln whar de brick was burnt,
But who throw'd de brick was nebber learn't.
 Chorus.--Bill Patterson rode bye &c.

I know ob a chap dat's up to de fun,
He knows who struck Bill Patterson;
But take my word he will nebber tell,
Unless somebody pays him well.
 Chorus.--Bill Patterson rode bye &c.

Dar's one ting sartin an plain for to see,
'Twas neider 'Sayres' nor 'Morissey;'
Dey both told me (or I is a liar,)
'Twas eider 'Heenan' or 'Old Tom Hyer.'
 Chorus.--Bill Patterson rode bye &c.

Money in de pocket shines so bright,
Old Bill got struck on Saturday night;
De lightnin flash__ he seen de 'seben stars!
He tink he was struck wid de bullgine cars!
 Chorus.--Bill Patterson rode bye &c.

If ebber you get to de 'Fiddler's Green,'
A labelled niggar can be seen,
Wid a sign on his back dat weighs a ton,
"I'm de darkie struck Bill Patterson!"
 Chorus.--Bill Patterson rode bye &c.

The song was a hit, appearing in numerous song books instantly. First performed in January 1860 by Emmett's troupe, Bryant's Minstrels, by April the song had been picked up by at least one other group, The Knights of Cork ([1], [2], [3]).

While the song as a whole remained popular with troops in the Civil War (with a detailed account of its singing by an Andersonville Prison Glee Club), the refrain was the most popular part among the soldiers; this song is what formed the nucleus of the the song I Can Whip the Scoundrel.

Riley Shepard, in his 1983 Master Book of American Folk Song, proposed that I Can Whip the Scoundrel originated from the slave song Pay Me My Money Down, as collected in the Georgia Sea Islands. The linked Mudcat thread reports the song being sung while unloading freight in 1858. I think it would be more accurate to say that "Pay Me My Money Down" was the basis of Billy Patterson, which then became "I Can Whip the Scoundrel."