The short answer is that Utah (and the Mormons) were hardly involved at all in the Civil War.
The Mormons of Utah Territory contributed a single company of cavalry, but their assignment was to protect the overland mail route and they were in service for only a few months. They never saw military action in the east. There were some individual Mormons who enlisted on their own and fought, but there were probably only about 2-300 of them.
Generally speaking, although the Mormons considered themselves a part of the United States, they were isolationists and some saw the Civil War as divine retribution upon the federal government for past wrongs. (The Utah War had been just a few years earlier in 1857-1858 so they were generally still distrustful of the Federal government at this point.) Additionally, this was also prior to the completion of the 1869 trans-continental railroad so traveling all the way from Utah Territory to the eastern US took months.
The most recent publication on the Civil War and Mormons that I know of is a collection of authored chapters called Civil War Saints. It was reviewed in detail at the Mormon history blog Juvenile Instructor. I haven't read the book myself, but they gave it a good review, and you might want to find a copy if you are interested in learning more about the subject. A good older book on this topic is The Saints and the Union: Utah Territory during the Civil War by E. B. Long (Urbana: University of Illinois Press, 1981)--it's a decent monograph. I think it might be an adapted version of his dissertation. If you're interested in learning more about the conflict between Mormons and the Federal government around this time period, I highly recommend The Mormon Question: Polygamy and Constitutional Conflict in Nineteenth Century America by Sarah Barringer Gordon (Chapel Hill: University of North Carolina Press, 2002).
Though I might not answer this question fully or give the answer you were looking for, it is interesting to note that Utah was one of the key sparkplugs for a larger sectionalist crisis. In the Compromise of 1850, Utah and New Mexico were allowed into the Union on the basis of popular sovereignty, which would ironically force the issue of slavery back into the public mind after the Second Party System had pushed it down for several decades. Therefore, the incorporation of Utah Territory in a formalized way after the Mexican-American War had key consequences that may have been factors causing the Civil War to break out.