I remember the close intertwining between the LDS Church and the John Birch Society when I was a kid. There were a few families that were Birchers, and seemed to take about it a lot. Even people that were just normal conservative Republicans didn't care for them. My cousins, at the house of the Bircher, were taken aside by the father of the family and given some lecture about communism.
I think here was a rule, written or unwritten, not to make a Bircher a Bishop. They thought Eisenhower and Nixon were commies.
In other words, a high percentage of Birchers were Mormons, but not necessarily vice versa. Ezra Taft Benson and David O. McKay, high leaders of the church, looked favorably on them.