Black Hawk, the leader of a band of Sauk and Fox Indians, fought against the United States Army and militia from Illinois and the Michigan Territory (present-day Wisconsin) for possession of lands in the area. This was called the Black Hawk War of 1832.
The Black Hawk War was responsible for the end of conflict between settlers and Indians in these states. One of the Illinois militia, Captain Abraham Lincoln, eventually rose to prominence in national politics becoming President of the United States.
There was also a Ute war chief in Utah named Black Hawk who, with a band of Ute, Paiute and Navajo, fought with the Mormons from 1865 until 1872.
The Black Hawk War was unique among the era's western Indian wars in that the antipathy that existed between the United States government and the LDS Church provided Utah's natives with the opportunity to pursue their hostile activities for an extended period of time without incurring the swift and destructive military reprisals suffered by other groups. Not surprisingly, the war ended almost without incident when federal troops were finally ordered to engage the Indians in 1872.
So, I guess in this case the UH-60 was named for an Indian leader. Which one, I do not know.