It's true that southern states pushed back against Brown v. Board of Education, and that federal law ultimately prevailed. On February 25, 1956, Virginia Senator Harry Byrd issued the call for "Massive Resistance"—a collection of laws passed in response to the Brown decision that aggressively tried to forestall and prevent school integration. Other states like Mississippi and Arkansas also implemented similar resistance measures, as documented in materials from the Legal Defense Fund. The Supreme Court reaffirmed its Brown v. Board decision following the Little Rock Nine incident in Cooper v. Aaron (1958), which maintained states' obligation to follow the Court's mandate to desegregate schools.
The original Bluesky post has been deleted, and it's unclear exactly which immigration-related cases the claim would be referring to, so I can't investigate the full context surrounding the "...government over immigration" point. However, the user was presumably asserting that federal law regarding immigration would similarly trump state law, just as it did in the integration cases.
