I have not been able to locate him in Confederate units, but, believe this may be him.

1860 United States Federal Census about William C Burford
Name: William C Burford
Age in 1860: 43
Birth Year: abt 1817
Birthplace: Alabama
Home in 1860: District 12, Shelby, Tennessee
Gender: Male
Post Office: Gum Grove
Household Members:
Name Age
William C Burford 43 Farmer
Elizabeth Burford 29 born in Tennessee
James Burford 8 -do-
Thomas Burford 2 -do-
John B Burford 20 -do-
Lucy Apperson 15 born in Mississippi
William Burford 4 -do-