Your Eli Branic may be the same as Eli Brannick shown on the 1860 US Census for McDonald Co. Missouri, age 15 (as of Jun 2, 1860), born in North Carolina. In 1850 they are in North Carolina -- Eli Brannock in Surry Co. NC, age 6. On March 1, 1875, Eli Branick (age 31, born NC moved to KS from MO) is living in Johnson Co. Kansas.
Eli Branic was enrolled in Benton Co. Ark. into company C of Bryan's Battalion Cherokee Partisan Rangers on July 4, 1862. In February 1863, Bryan's Battalion was consolidated with additional companies from the 1st Cherokee Mounted Volunteers to form the 2nd Cherokee Mtd Vols commanded by Col. William Penn Adair. On the roll of June 30, 1863, Branic is on detached service with Lt. Hendrin "on scout with Major Livingston" since April 27, 1863.