In a back room at the Franciscan House of Mary and Joseph, one of the largest homeless shelters in Chicago, a social worker named Sheena Ward guided Terry Cannon through a Medicaid application. A wet cough punctuated Mr. Cannon's often wry answers to Ms. Ward's questions about his disability status, military service and marital history. "I have glaucoma, I'm going blind. I have lung disease, I'm dying," he said. "How can they deny me? If they do, give me a couple years and I'll be gone." Today, most state Medicaid programs cover only disabled adults or those with dependents, so Mr. Cannon and millions of other deeply impoverished Americans are left without access to the program.