Yes, people made the argument equating the two. The logic ran: the NAACP (which brought up the Brown case which caused school desegregation) was full of communists (they weren't, this is the fervent segregationists talking) and so their support of desgregation means that "race mixing is Communist".

The NAACP had suffered under accusations of Communism since the start of the Cold War, and tried to make a firm effort to separate themselves; at their 1950 Boston Convention, they made a resolution that allowed the national office to expel any "communist-infiltrated" unit; while they weren't actually Red at the time (so to speak) they were sending a message.

The message didn't work, and segregationists sincerely believed the NAACP was "communistic"; this was essentially mythos building at a deep, almost religious level. The best example of this is Eugene Cook's pamphlet "The Ugly Truth About the NAACP" which was the most widely read text about the subject. It was originally a speech to the Peace Officers' Association of Georgia.

The files of the House Un-American Activities Committee reveal records of affiliation with or participation in Communist, Communist-front, fellow-traveling or subversive organizations or activities on the part of the following present officials of the NAACP -- the President, the Chairman of the Board, the "Honorary Chairman," 11 of 28 Vice Presidents, the Treasurer, 28 of 47 Directors, the Chairman of the National Legal Committee, the Executive Secretary, the Special Counsel, the Southeast Regional secretary, the West Coast Secretary, the Director of the Washington Bureau, the Director of Public Relations and two Field Secretaries.

The NAACP was generally but not universally equated with communism. A short book by a former leftist, J.B. Matthews, entitled Communists, Negroes, and Integration, made special note of the NAACP not being a "Communist Front" but still claimed that desegregation was part of the Communist strategy to cause "racial termoil". This hits upon the other argument in the Eugene Cook text; while the majority of the pamphlet is filled to the brim with conspiracy allegations about members of the NAACP, near the end it tries to form something resembling an argument: that the NAACP, as part of their Communist front activities, is "fomenting strife and discord" and disrupting "harmonious and friendly" relations. Essentially, by their logic, the so-called "race mixing" is causing strife, and disrupting harmony is what the Communists want so hence "race mixing is Communist".

You can read the pamphlet in question here via the University of Mississippi Commons.