On a frigid morning here, Nancy Godinez was piling bread and other staples into her car outside a food pantry. She had lost her job as a custodian, her unemployment checks had run out, and her job search had proved fruitless.

One thing she still had was health insurance, acquired three years ago after Arkansas’ Republican-controlled legislature agreed to expand Medicaid under the Affordable Care Act. The coverage, she said, has allowed her to get regular checkups and treatment for tendinitis in her foot.

But unless she finds a new job, Ms. Godinez, 55, could be at risk of losing her insurance, too.