TONGANOXIE, Kan.—Rural Americans are turning their backs on the industry that made the U.S. the biggest meat-exporting country in the world.

Residents of Tonganoxie, a 5,300-person town in northeast Kansas, spent part of the fall hanging white-and-red placards that say “No Tyson in Tongie” on fenceposts and pickup trucks. Their efforts were part of a public push against Tyson Foods Inc., the largest U.S. meat processor by sales, which trumpeted in early September its plans to build a $320 million chicken-processing complex...