I'm not related, just trying to help.

I think you're on the right track. I've looked but can't find them in 1860 or 1870. It looks like they should be living in Lawrence Co, MO during that time. Here are 3 marriage records from Lawrence Co for 3 of the Shultz children including Robert.

Missouri Marriage Records, 1805-2002
Name: Robert Shults
Marriage Date: 1 Mar 1863
Marriage County: Lawrence
Spouse Name: Martha Jane Strohm

Missouri Marriage Records, 1805-2002
Name: Julia Ann Shults
Marriage Date: 8 Feb 1865
Marriage County: Lawrence
Spouse Name: Daniel Pennington

Missouri Marriage Records, 1805-2002
Name: Mary Shultz
Marriage Date: 7 Jul 1860
Marriage County: Lawrence
Spouse Name: Andrew Morrison

And here's the military record for Wiley Shultz. It shows he enlisted in Lawrence Co, MO in 1862.

U.S. Civil War Soldier Records and Profiles
Name: Wiley Shults
Age at enlistment: 16
Enlistment Date: 11 Jan 1862
Rank at enlistment: Private
Enlistment Place: Lawrence County, MO
State Served: Arkansas
Survived the War?: Yes
Service Record: Enlisted in Company I, Arkansas 1st Cavalry Regiment on 01 Nov 1862.
Birth Date: abt 1846
Death Date: 1 Apr 1902
Death Place: Siloam Springs, AR
Sources: Index to Compiled Military Service Records
1st Arkansas Union Cavalry - Arkansas Research