Generally speaking, most insurance premiums cannot be covered by an HSA; they are not included on the eligible expense list published by the IRS. However, there are a few exceptions. According to IRS publication 969:

Insurance premiums.

You can’t treat insurance premiums as qualified medical expenses unless the premiums are for:

  1. Long-term care insurance.
  2. Health care continuation coverage (such as coverage under COBRA).
  3. Health care coverage while receiving unemployment compensation under federal or state law.
  4. Medicare and other health care coverage if you were 65 or older (other than premiums for a Medicare supplemental policy, such as Medigap). “

Yes, COBRA premiums can be paid out of your HSA and you can continue coverage. In addition, if you are unemployed and are receiving unemployment compensation, you can purchase other healthcare insurance with your HSA.