[the following is opinion written by an experienced ESL teacher in China – NOT by a physician – seek competent medical guidance for your personal health issues]
I have noticed that the Chinese really care about health of their nation. For this reason, all newly married couples are required to pass medical checks and figure out if they are able to give birth to new healthy generation. If not, they have to take necessary treatment to avoid health problems.
Also, for this reason every foreigner, who attempts to get a working visa, has to pass medical checks as well, otherwise you won’t get your working visa in China at all.
I have been in a few places in China where I had to pass medical checks, and I have noticed that hospitals are clean in most cases and provide a good service and medical care. I think the situation with medical care is far better in large cities with populations of 3-15 million people. In such cities you can even find private clinics of Chinese traditional medicine with English speaking doctors and nurses. But I doubt that such visits to private clinics would be covered by your health insurance.
Before signing a contract check if your employer offers you a package that also includes medical insurance as a part of your benefits package for that school. Employers sometimes offer health insurance and they might pay for it without deducting any money from your salary. Insurance coverage is not expensive in China, since Chinese medicine is very cheap.
There are a few different health insurance plans in China. Basically these plans cover cases of emergency, doctor visits, hospitalization, surgery, prescriptions and dental needs, but do not usually cover cosmetic surgery or alternative medicines, infertility issues, or pre-existing conditions. Also, injuries from extreme sports or activities are not normally covered. It is advisable that you ask your employer about insurance, and if you are not satisfied with the package you can buy your own health insurance.