If you apply in your home country for a non-immigrant visa, you are applying for an OA visa.  To get an OA visa you must show proof of health insurance.  It would be a good idea to maintain the insurance you now have, assuming your insurance provides coverage in Thailand.

If you apply for a non-immigrant visa by converting your tourist visa in Thailand there is no proof of insurance required at this time.  However, in all likelihood, it may become a requirement that all expats have health insurance.  If that does happen and you need to buy insurance locally you will probably have no coverage for pre-existing conditions and depending on your age it could cost you $3000.00 plus per year.

The minimum coverage requirements are 400,000 baht for inpatient and 40,000 baht outpatient.