I know that the north western part (what became West Virginia) of Virginia and many border states were indifferent to the Confederacy. But what about regions in the deep south? Did any counties or towns refuse to help the Southern cause?
East Tennessee was maybe the largest, most unionist part of the south below the 36th parallel. East Tennesseean insurgents burned bridges, confederate soldiers hanged east Tennesseeans. There's a book in a Baptist Church in Cades' cove in which someone wrote that they have not me in the church during the war to avoid harassment because 'we was union men'.
Source: Battle Cry of Freedom, McPherson
The Acadian regions of Louisiana had no interest in the war. They hid out in the swamps and took on all comers.