Prunes are known to be nature’s laxatives. The bumper sticker “Start A Movement—Eat A Prune” (a joke on “bowel movement”) has been cited in print since at least 1973.
   
   
Wikipedia: Prune
Health benefits
Prunes and their juice contain mild laxatives including phenolic compounds (mainly as neochlorogenic acids and chlorogenic acids) and sorbitol. Prunes also contain dietary fiber (about 6%, or 0.06 g per gram of prune). Prunes and prune juice are thus common home remedies for constipation. Prunes also have a high antioxidant content.
 
4 February 1973, The Sun (Baltimore, MD), “State’s legislators face a social 2 weeks,” pg. A14:
“It read: ‘Start A Movement, Eat A Prune.’”
 
18 August 1973, Nevada State Journal (Reno, NV), pg. 5, col. 1:
Two bumper stickers sighted this week impart profound messages:
 
Support Mental Health—or I;ll Kill You;
And: Start a Movement—Eat a Prune.
 
19 May 1974, The Independent Record (Helena, MT), “A bumper crop of stickers,” pg. 17:
START A MOVEMENT
EAT A PRUNE
 
Google Books
Reporting for the Print Media
By Fred Fedler
Fort Worth, TX: Harcourt Brace Jovanovich
1993
Pg. 401:
Another student appeared for an interview dressed in a T-shirt inscribed, “Start a Movement—-Eat a Prune.”
 
Google Books
Only Begotten Daughter
By James Morrow
San Diego, CA: Harcourt Brace
1996
Pg. 149:
At least she was dressed for the heat: denim cutoffs, a Smile Shop T-shirt (start a movement — eat a prune).
 
Google Books
The Mammoth Book of Humor
Edited by Geoff Tibballs
New York, NY: Carroll & Graf
2000
Pg. 331:
Eat a prune and start a movement.