Such an assertion proceeds from a false premise. The secessionist states were acting constitutionally, not rebelliously. Consequently, there could be no insurrection on their part, only acts of self-defense.
It was northern aggression directed by the Federal Union in the person of Lincoln that acted to crush the secessionist states for seceding, something the states had every right to do. Lincoln by force of arms would compel the states to rejoin the Union, or they would be destroyed. There can be no doubt that the Federal Union broke the compact. The secessionist states wanted only to leave the Union in peace and go their own way, independent, sovereign, and free. As was their right.
The original Constitution as agreed to by the several states specifically enumerates the powers granted to the Union government and explicitly states that all powers not granted to the Union government are reserved to the states, or the people. Since the authority to remove a state from the Union is not granted to the government, (i.e. Congress can't remove Virginia) the authority is reserved to the states.
The 10th amendment was never more clear or pertinent. The ultimate authority of deciding if a state is to stay with or vacate the Union was not granted to the Union, thus it resided solely with the states.
That's it, that's all it ever was. Lincoln could not have cared less. He couldn't wait to violate the compact and ignore his oath of office. So he didn't.
Thank you for using the proper plural form. The only way to secede was to do so mutually - by the states. That wasn't how the cornfederates did it. They chose unilateral secession which wasn't recognized by anyone except themselves.
Lincoln honored his oath by putting down an insurrection.
But if we accept your premise that the southern states had left the union, as you say was their right, then there was no more compact for Lincoln to break. You can't renounce your place in the union and under the Constiution then complain that they no longer protected you.