Healthcare in Vietnam operates on a two-tier system of both public and private health insurance coverage, and has been improving in recent years, but it is unlikely that, as an expat, you will be able to access the public health insurance scheme. You will therefore need to take out private health cover.
Although the principle of universal coverage has been growing in the country, it is still patchy, particularly in rural areas, and in any case you are unlikely to be eligible for the national insurance scheme. Healthcare itself suffers from underfunding and structural problems. You can still access public healthcare itself, but will need to pay some out of pocket payments. Moreover, your employer may either sign you up with a private policy as part of your employment package (check the coverage on the policy, however) or may give you the money to arrange your own health insurance provision.
Note that you will still need to pay social security contributions via a monthly payroll deduction, despite your lack of access to public healthcare. At present this is around 8% of your salary.
How to register for healthcare in Vietnam
If your health insurance provider covers Vietnam, consult them for a list of approved private medical providers. If you wish to register with your local clinic, check online or with the expat community. In Ho Chi Minh City and Hanoi, the French Hospitals have an excellent reputation: these are government hospitals and you will need to pay for treatment directly, but as we note below, this is usually inexpensive.
You can find a list of hospital medical personnel here.
You will need to pay out of pocket for state medical care in Vietnam but this is usually low cost. For example, an X-ray is in the region of USD$1.50 and an ultrasound is under USD$5.
As an expat, your dental treatment is likely to be within the private sector. However, as a destination for dental tourism, Vietnam is very competitively priced, particularly in comparison with the USA. Treatments that could cost you around USD$3K will be available in Vietnam for under USD$1K, for example. There are a large number of private clinics, particularly in urban areas: it is advisable to shop around, compare prices, and consult your local network, especially if you are planning to undertake cosmetic procedures.