During desegregation, many whites claimed that their rights were being infringed upon by forcing them to interact/do business with blacks. How did the government get past this argument?
The Supreme Court case of Heart of Atlanta Motel vs the United States established that Congress could force private business to serve blacks under the Commerce Clause. The Court held that the 5th Amendment protection of private property was subject to reasonable regulation by Congress.
People could still have racist private clubs, but once it engages in commercial activities the Civil Rights Act applied. Under the Courts previous decisions, the definition of 'interstate commerce' was extremely broad.