EJIDO MODELO, Mexico — On the two-hour bus rides from her village on Lake Chapala to a dialysis clinic in Guadalajara, Monica Chavarria’s thoughts would inevitably turn to the husband and son she left behind in Georgia. A decade after crossing illegally into the United States, Ms. Chavarria returned home in September after learning that Grady Memorial Hospital in Atlanta was closing the clinic that had provided her with dialysis, at taxpayer expense, for more than a year. Grady, a struggling charity hospital, had been absorbing multimillion-dollar losses for years because the dialysis clinic primarily served illegal immigrants who were...