Lincoln took the position of King George the III, and the Southern States took the official position of the US government as outlined in it's founding document.
That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, --That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness.
Now if you want to take issue with what rights they exercised, you should take it up with the Founders of the US government who not only recognized those rights, but who wrote protections for them into the US Constitution.
So they left because they thought Lincoln would interfere with their right to own slaves. Where in the Constitution is slavery a “protected right”