This term gets thrown around a lot, and as much as I rally against the overuse of it, I want to clarify that this is Alien Space Bats of the highest form.
The South will not seceded over the Civil Rights Act. No sane politician in the South during this period will seriously propose the reformation of the Confederacy, and no sane state would ever get so carried away that they would try independence. Even if someone got carried away, the core component of the Lost Cause mentality was recognizing that the South was doomed to fail, and Lost Causers new that the South was as doomed to failure in 1964 as they were in 1861.
George Wallace, Storm Thurmond, Robert Byrd, Richard Russell, and the rest of the Southern Bloc are not going to respond with secession when they can fight it through legislative and political means. Secession is insane. It is madness. It is so out of character for every single one of these people who would be involved that you might as well ask if they would arrange to put golf balls at the white house.
The bottom line is a flat no. The South will not seceded, and the CSA will not reform over the Civil Rights Act. But lets assume that somehow, just somehow, this happens.
What next? The nukes don't belong to those states, they belong to the Federal Government. The men in the bases did not swear an oath to the CSA, they did so to the USA. So what happens? The conspirators of this secession are immediately ousted and arrested by the Federal Government. Martial Law is declared in these states, with the Army and National Guard patrolling the streets and enforcing curfews. The US takes back control within a fortnight at the latest. The Civil Rights Act gets passed.
But this won't happen. This won't happen simply because no one who was representing their state in the House, Senate, or as a Governor, was stupid enough to suggest that they seceded from the people who literally pay their bills and could crush any unilateral and armed uprising like a roach beneath a boot, over a legislative Act that they could resist more effectively in their roles in the House, Senate, and Governorship's. Even if this was post-Civil Rights Act, it's still madness to propose this, and even more nonsensical. They failed in their opposition to an Act they didn't like, but they can still further opposes it and screw with Johnson's Civil-Rights Agenda through other means that does not end with them in from of a judge, reputation ruined and facing charges of Treason.