Walter Reuther
Walter Reuther was a big shot in unions. He led the UAW (United Auto Workers) and the CIO (Congress of Industrial Organizations). During his time as President of the CIO, he negotiated its merger with the AFL (American Federation of Laborers). He worked for Ford as a tool and die maker until his leaving the company. From 1933 to 1935, he and his brothers worked in Russia. Reuther was a socialist, but never became a communist, despite his time in Russia. He sometimes cooperated with communists, when their goals aligned. He was not fond of Stalin. Reuther was also a member of the Intercollegiate Socialist Society (a predecessor to the League for Industrial Democracy), heading its chapter at Wayne State University.
According to the AFL-CIO's website, Reuther met weekly with President Lyndon B. Johnson on legislative and political issues between 1964 and 1965. By coincidence, Medicare was created by the Social Security Amendments of 1965. In this way, I suppose he's Johnson's Stern or Trumka.
Speech Relevance
In 'Encroaching Control':
Walter Reuther has announced that it is no secret that the organization that he represents favors a complete program of national health insurance for all the population.
In 1964, Reuther was head of the UAW, so we must assume this was the organization Reagan spoke of.
In 'Ronald Reagan Speaks Out Against Socialized Medicine' we hear the actual quote:
"It's no secret that the United Automobile Workers is officially on record of backing a program of national health insurance."
The source of this quote is unknown, even to a UAW-specific archivist at the Walter P. Reuther Library of Wayne State University.