No states currently offer their own healthcare plan for all their citizens.  However, one state had their own system before Obamacare.

Massachusetts

The state built a precursor to the ACA, “RomneyCare,” in 2006.  The program resulted in 97% of Massachusetts residents having healthcare.  The program ended with the advent of the ACA.

In the rest of the world, the United States and Mexico are the only countries without universal healthcare:

Countries with Universal Health Care are in Green