What are the insurance requirements for a Non-O visa application in Thailand, and what recommendations do you have for insurance providers?

There is no such thing as a 12 month non-O multiple entry visa.

You have to get a 90 day non-O visa, which is single entry. Then once you are in Thailand you open a bank account, transfer the required funds (800,000 baht) within 30 days of arriving to Thailand, and then after the money has been in the bank for 2 months, you can apply for the 1-year extension (not a visa anymore, an extension).

Once you're on the 1-year extension, you can apply for a multiple re-entry permit. That does not make your extension multiple entry, it makes it multiple re-entry. Multiple entry is a completely different thing.

What a re-entry permit does is keep your extension alive if you leave Thailand. So normally if you left without one, your extension is cancelled as soon as you leave, and when you came back you would have to return as a tourist and start all over.

With a re-entry permit, your stamp stays alive, but you do not gain any extra time, and the time keeps ticking while you are out of Thailand.

So if your extension ends July 30, and you leave on March 30 with a re-entry permit, and then return on April 30, you'll be stamped back into Thailand until July 30th. If you're not back in Thailand before July 30th to apply for your next extension, you would lose it and have to start over.

The non-O visa does not have an insurance requirement. You are free to choose any insurance you want.