Signing up for insurance on the new health-care marketplaces can be challenging enough without having to remember your high-school mascot. Or your favorite childhood superhero, the manager at your first job or the street you lived on in third grade. Those are just some of the security questions being offered to shoppers on the state and federal exchange sites as a way to verify their identities—and they are leaving some would-be customers flummoxed. "I don't think they took baby boomers into account when they invented those questions," says Margo Benge, a 58-year-old massage therapist in Missouri City, Texas, who gave...