From the Wikipedia web page regarding the origin of the Jayhawk:
The origin of the term "Jayhawk" (short for "Jayhawker") is uncertain. The term was adopted as a nickname by a group of emigrants traveling to California in 1849. The origin of the term may go back as far as the Revolutionary War, when it was reportedly used to describe a group associated with American patriot John Jay.
The term became part of the lexicon of the Missouri-Kansas border in about 1858, during the Kansas territorial period. The term was used to describe militant bands nominally associated with the free-state cause. One early Kansas history contained this succinct characterization of the jayhawkers:
“Confederated at first for defense against pro-slavery outrages, but ultimately falling more or less completely into the vocation of robbers and assassins, they have received the name --- whatever its origin may be -- of jayhawkers.”
Another historian of the territorial period described the jayhawkers as bands of men that were willing to fight, kill, and rob for a variety of motives that included defense against pro-slavery "Border Ruffians", abolition, driving pro-slavery settlers from their claims of land, revenge, and/or plunder and personal profit.
In September 1861, the town of Osceola, Missouri, burned to the ground by Jayhawkers during the Sacking of Osceola, asked the University of Kansas to remove the Jayhawk as its mascot.
Over time, proud of their state's contributions to the end of slavery and the preservation of the Union, Kansans embraced the "Jayhawker" term. The term came to be applied to people or items related to Kansas. When the University of Kansas fielded their first football team in 1890, the team was called the Jayhawkers. Over time, the name was gradually supplanted by its shorter variant, and KU’s sports teams are now almost exclusively known as the Jayhawks. The Jayhawk appears in several Kansas cheers, most notably, the "Rock Chalk, Jayhawk" chant in unison before and during games. In the traditions promoted by KU, the jayhawk is said to be a combination of two birds, “the blue jay, a noisy, quarrelsome thing known to rob other nests, and the sparrow hawk, a stealthy hunter.”
The link between the term “Jayhawkers” and any specific kind of mythical bird, if it ever existed, had been lost or at least obscured by the time KU’s bird mascot was invented in 1912. The originator of the bird mascot, Henry Maloy, struggled for over two years to create a pictorial symbol for the team, until hitting upon the bird idea. As explained by Mr. Maloy, “the term ‘jayhawk’ in the school yell was a verb and the term ‘jayhawkers’ was the noun.” KU’s current Jayhawk tradition largely springs from Frank W. Blackmar, a KU professor. In his 1926 address on the origin of the Jayhawk, Blackmar specifically referenced the blue jay and sparrow hawk. Blackmar’s address served to soften the link between KU’s athletic team moniker and the Jayhawkers of the Kansas territorial period, and helped explain the relatively recently invented Jayhawk pictorial symbol with a myth that appears to have been of even more recent fabrication.