Connie, the key to tracking down the Confederate service of a man with a very common surname, such as Smith or Johnson, is to determine where he was living when the war started (use the 1860 census). Typically, Arkansas men tended to join companies organized nearest their homes. If you find a William Johnson in such a company, check the names of the other soldiers in that company against the county's 1860 census. If those names are neighbors and/or relatives of your William Johnson, you can be pretty sure that he's the soldier you're looking for.
On the other hand, your ancestor may have served in the Union army. I found the record of a William Johnson who died at Big Flat on October 5, 1912. He was born c1835 in Schuyler County, Illinois, and served in the Union army. He enlisted in Co. E, 1st Arkansas Cavalry, U.S. Volunteers, at Cassville, Missouri, on June 19, 1863, for three years; age 28. He applied for a U.S. veteran's pension in 1892.
If this soldier is your ancestor, you can order a copy of his service and pension records from the Military Records Service linked to this message board.