In Topeka, many African American teachers at the segregated schools were against integration, because they knew the integrated schools wouldn't hire them as teachers.

http://law2.umkc.edu/faculty/projects/ftrials/brownvboard/brownaccount.html

Topeka almost had to be where the lawsuit took place. It didn't just happen over a year or two, but the fights for integration/segregation took place over decades with many in the community wanting full integration, and many of the high schools were integrated (although not all extracurricular activities were integrated). It meant that the community wasn't deeply entrenched in all levels like other towns and cities had been.

" The contingent of twenty-five educators and parents including the recently retired principal of the all-black Washington School Ezekiel Ridley, declared that “at least 90 percent of the colored people in Topeka want their children to go to the colored schools” and requested the board continue to maintain grades seven and eight in the city’s four black schools....Resolving to wage an “unrelenting fight in behalf of the children” the attorney [former president of the Topeka NAACP and attorney Raymond Reynolds] and his committee condemned the plan offered by their opponents as being designed only to “save a few colored teachers their jobs at a sacrifice of the rights assured to the children under the court’s decision.”

http://www.nps.gov/brvb/historyculture/topekasegregation.htm

Yes, this is the lead up to Brown, but I wanted to leave this information anyway. I'll delete it if asked.

So we have Black Power:

1- Nation of Islam (founded 1930), a group dedicated to improving the social, economic and political standards of African Americans. Lots of people labels these guys as black separatists/ black nationalists/ pan- africanists due to the Elijah Muhammed (the leader) and Malcolm x publically speaking about how African Americans needing "pride in their own race" and wanting a consolidation settlement from the USA as they wanted a separate country from "the white devil" due to their treatment of black people.

Malcom X especially spoke out against integration is his earlier years, he spoke out repeatedly on Martin Luther King only wanting integration as MLK was a "puppet" for the US government and that anyone to follow MLK was a "sheep following the shepherd"

2- Black Panther Party (founded 1966), just to make this clear this group are nationalists so are not necessarily separatists but they are notable in voicing a lot of what the separatists believed in such as helping their own race, notifying them on their legal rights and providing social programs.

3- Individuals wanting separation of the whites and blacks can be Marcus Garvey, he wanted those with African ancestry to leave America and go back get rid of European colonizers in Africa.

Red Power:

1- American Indian Movement (founded 1968) who were heavily influenced by the Black Power Movement so they pretty much had the same aims and methods as the black separatists. But some extra info on why these guys were separatists, so Native Americans (NA) were being torn apart by the US government day-to-day with plans like the Indian Termination Policy 1953-64 which basically made Native Americans lose their reservations, exclude them from state laws and terminate the recognition of tribal sovereignty, and crazy loads of racist stereotypes etc. Obviously this made them mad and want change, at the time (1960s-70s) the Black Power Movement were becoming popular and were making headway in the Civil Rights Movements so the NA took their methods of gaining equality into their own hands.

Source: Nation of Islam information, Malcolm X's The Ballot of the Bullet, Ward Churchill's Agents of Repression: The FBI's Secret War Against the Black Panther Party and the American Indian Movement and Gillian Ronson's The American Indian Struggle.

This doesn't fit your criteria exactly, but you'll probably be interested in the 1968 NYC teachers strike.

The black people of Brownsville weren't against Integration per se, but the Black students who went to White schools reported heavy mistreatment, and they figured that integration would never happen (they were right, New York schools never integrated to any appreciable degree, segregation actually increased post-Brown), so instead they tried to institute community control of their schools. It was working out pretty well (except for the white teachers who got fired), But it was quickly squelched and now Brownsville has some of the worst schools in the nation

YES: there were indeed "notable black or minority groups who ... spoke out ... against integration ..." While there is an overwhelming literature on Brown v. Board of Education (I assume your "Board America" was a typographic accident), Zora Neale Hurston's letter to the editor of 1955 was the most ... notorious direct and contemporary rebuke to this milestone of desegregation.

Others nearby have already mentioned Black Panthers, the Nation of Islam, the American Indian Movement, ...: there were also complex local calculations about the impact of Brown v Board of Education in multilingual districts. Also note that current Justice Clarence Thomas has prominently demurred from common interpretations of Brown vs Board of Education.

Moreover, I think it's important to note that Brown v Board of Education was only one in a sequence of court rulings that bore on educational segregation; several of the earlier cases had featured other minorities (students of Chinese, American Indian, ... descent) along with blacks.