Ezra Taft Benson was, as far as I know, never a member of the JBS.  His wife was an active member until she died, and his son Reed worked for the JBS for many years.

As an Apostle, Ezra Taft Benson wrote in private correspondence to a Brother and Sister Curtis in Manti, UT: "You will be interested to know that President McKay encouraged my son, Reed, to stay with the John Birch Society indicating that it would be vindicated and become a powerful organization in the preservation of our freedom."

And in another private letter to a Brother Olsen in Spokane, WA he wrote: "The John Birch Society has been smeared and misrepresented by the communist and liberal press in this country. I consider the Birch Society the most effective secular organization in America in the fight against socialism and godless communism."

(http://www.godfamilycountry.net/davidomckayandbirch.html)

The John Birch Society promotes the principles expressed in the Declaration of Independence and codified by the U.S. Constitution.  Its goal includes persuading Americans to find and support candidates for public office who will promote the return of government to its proper constitutional levels.  And LDS Church leaders have, more so in the past than recently, urged members to get involve in supporting such efforts.  The JBS also promotes basically the same family values and moral principles as the LDS Church.  In spite of this, it is still smeared and misrepresented even by Church members.

Ezra Taft Benson stated:  "For years we have heard of the role the elders could play in saving the Constitution from total destruction.  But how can the elders be expected to save it if they have not studied it and are not sure if it is being destroyed or what is destroying it?"   He also wrote "...the last neutralizer that the devil uses most effectively - is simply this;  "Don't do anything in the fight for freedom until the church sets up its own specific program to save the Constitution.  This brings us right back to the scripture I opened with today - to those slothful servants who will not do anything until they are "compelled in all things."   Maybe the Lord will never set up a specific Church program for the purpose of saving the Constitution.   Perhaps if he did it would split the church asunder, and perhaps He does not want this to happen yet, for not all the wheat and tares are fully ripe." (Teachings of Ezra Taft Benson)