The war of 1861065 was a revolution, which is why one name for the war was "the War for Southern Independence." Pointless to talk about rights under the Constutution when the Confederates were breaking away from it. The matter was not to be settled by argument but by force of arms, just as in 1776. Despite union victories, the South came close to independence. If John Hood had made a more intelligent defense of Atlanta and held the Union forces off for another six week's Lincoln would have been defeated for re-election, and McClellan led a party that accepted disunion.
Well, if the South was in Revolution, then it needed to justify that revolt with an appeal to an attack on its individual rights.
The Federal gov't was not abusing anyone's individual rights in the South.
As for the war being settled by force of arms, if the Colonists had lost the war that would not negated the rightness of their cause, just as if the South had won, it would not have justified their cause.
Fortunatly, the winners in both were those who were defending the just cause.