GABRIELS, N.Y. — After 17 years of marriage, Joy and Richard Gonyea managed to save enough to trade their trailer in November for a cozy prefabricated home with a room for each of their two children and a pool in the backyard. The home overlooks the pine trees on the edge of their two-acre property in rural Vermontville, eight miles from the secluded state prison where Mrs. Gonyea works. “This home is all we’ve ever dreamed of,” said Mrs. Gonyea, 43, a registered nurse who runs the medical department at the prison, Camp Gabriels, a minimum-security facility in this minuscule...