States rights (the right of a state to leave the Union) became a subject of vigorous debate some years after Appomattox. Southern citizens carefully following the election of 1860 and the ensuing secession crisis would not have read one editorial on states rights. Prior to Fort Sumter, most Americans presumed that each state possessed the right to leave the Union. However, many people (including Robert E. Lee) believed it highly unwise to resort to secession.
Breckinridge-Lane supporters were frequently called States Rights Democrats, but the reference had nothing to do with secession. During the 1850s States Rights meant the right of a citizen from any state to hold slave property in a U.S. territory. That argument split the Democratic Party in 1860. Senator Stephen Douglas, the national Democratic nominee, supported the right of a territory such as Kansas to restrict or exclude slavery by a vote of the citizens. As an advocate of 'Squatters Rights', Douglas was reviled in many quarters of the South, and John C. Breckinridge nominated to oppose and defeat him.
Just so you know, opponents of the Breckinridge-Lane ticket accused these gentlemen of advocating secession, a charge which they vigorously denied. You may think of it like modern-day charges of racism against political leaders -- it's frequently done to smear a candidate, and always denied.