The relevant text of the 14th Amendment here is its opening declaration: "All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside. No state shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States ...." [0] (Emphasis added.)
In other words: Under the 14th Amendment, United States citizenship takes precedence over state citizenship — kind of like the old boast, civis romanus sum: I'm a Roman citizen. (The implication was: fuck with me at your peril.) Cf. Act 22:25: "As they stretched him out to flog him, Paul said to the centurion standing there, 'Is it legal for you to flog a Roman citizen who hasn’t even been found guilty?'"
To be sure, there are Lost-Cause diehards who claim that the 14th Amendment never actually came into effect because the Southern states refused to ratify it and so it didn't get the necessary 3/4 of states; courts have consistently rejected that claim [1]. If anything, the former Confederate states should be grateful that they didn't get the treatment that the Allies meted out to the Germans and Japanese in 1945 and afterwards — one recent commentator, remarking on the outrages perpetrated in and by the post-Civil War South, tweeted that "Sherman should've mowed the deep south like a lawn, making multiple passes." [2]
[0] https://www.law.cornell.edu/constitution/amendmentxiv
[1] https://www.splcenter.org/fighting-hate/intelligence-report/...
[2] https://twitter.com/pptsapper/status/1313470161974947843