What are the recommended companies for compulsory healthcare insurance for the non-OA visa in Thailand?

Use AXA

It also might be worthwhile mentioning that depending on your age Health insurance in Thailand can be extremely expensive. Check out the costs before you choose what visa to get.

Are you sure you actually want the non-OA visa? That's the only one that requires health insurance, and it's not the retirement visa most people want. Most people want the non-O visa and plan to get annual extensions inside Thailand. This visa has no insurance requirement and doesn't require the other things like background check, health check, etc.