BOONE, N.C.—Eustace Conway says he has stared down a grizzly bear, wrestled a thrashing buck and ridden a horse from coast to coast. But he may have met his match in the Watauga County planning department.

Mr. Conway, 51 years old, is best known as “The Last American Man,” the title character of a 2002 biography and National Book Award finalist by Elizabeth Gilbert, the author of “Eat, Pray, Love.” He has lived…