That really depends on the embassy you apply with, I have medicare and v.a insurance; neither one of those worked I have to buy a policy they approved of, but honestly I don't it, I still have coverage from my employer and that did not work either.
I have the LTR visa WP) applied at the embassy in Washington D.C. Medicare does not cover, you will definitely have to get the Health insurance that Thailand approves of, or have $100,000.00 cash in your account for 12 months consecutively
I went through an agent for mine and got my policy within 5 minutes via email delivery. I then had to upload the entire policy, including the 36 pages to BOI for the LTR(WP) visa