In order to accept HSA and FSA cards using Square, your business will need to be set up under the business category Medical Practitioner, Medical Services, or Dentistry in Square. This can be selected during Square account creation. If your account has already been created, reach out to Square Support with your request, here.
Note for previously connected Stripe accounts:
All Stripe accounts (if created through MassageBook) are automatically classified under MCC 8099. Since Square accounts are not created through MassageBook, we can't do anything on our end to make that card accepted by Square. You will need to contact their Square Support to ensure you can accept HSA and FSA cards of a certain classification. Your Stripe account still works regardless if its connected through MassageBook, so for an immediate solution to process those while still being connected to Square, you can log into your Stripe account via stripe.com and try to charge the card.