I assume you mean the Covid19 health insurance? General health insurance is always a good thing too but you don't need it to get a non-O visa if that is what you are planning to do.
So that means the OP’s friend has to decide between an O-A obtained in the US or to come to a Thailand and get a non-O in a neighboring country or inside Thailand. That is an important distinction.
Well, to start, an O-A visa is going to require health insurance. For non-Thai insurance you need your insurance provider to sign a document saying your coverage meets the requirements. Otherwise you need insurance from an authorized Thai companies. In general, is an insurance requirement going to be a non-starter for you?
Can’t you just change the reason for your extension the next time you apply?
If you need to do it sooner, and you don’t have a multiple re-entry permit then you can just leave the country and your extension is void. You can return to Thailand on a visa exempt and apply for an O inside Thailand or get one out the country. Different rules apply.
If you intend to stay on a non-OA then the next time you apply in your home country you will either need Thai insurance or have your outside insurance company sign the Thai government affidavit. If you apply for a 1 year extension of stay based on your non-OA then you will need a Thai insurance. Robert answered what might happen to you currently.