Hi Sean, and sorry for the long delay in answering this question.
It’s true that some of the accomplishments of the Civil Rights Movement came in the form of legislation that mandated racial integration on private property. This is a problem from a libertarian perspective because it impinges on the rights of property owners. Figures like Barry Goldwater protested the Civil Rights Act even though he had already desegregated his businesses.
So the standard libertarian response to this was to favor abolition of the state laws that mandated segregation in public and/or private spaces, and oppose laws that mandated racial integration.