No they don't. They have state provided health care, supplemented with private options.
But they are not universal in the sense that (1) not the entire population is covered (although almost everyone is) (2) health care is not close to free for most people.
https://en.wikipedia.org/wiki/Universal_health_care_by_country#China
There is literally a hit movie couple years ago about the difficulty to obtain affordable Imatinib: https://en.wikipedia.org/wiki/Dying_to_Survive
(Imatinib was later covered by state health care package because of this movie