Not really. The problem with health care is that it works if it's socialized, and it doesn't if it's private, as shown by about all the entire developed world. And Saudi Arabia. No way to patch it to make it palatable to the right wing, but that shouldn't be a problem. There wasn't any way to make the USSR being a failure palatable to the left, either.
To clarify, health care is not a consumer product, it's a particular kind of insurance. It doesn't make sense to bring consumer choice into it. [1]
> if you are left-leaning, maybe you can support a universal minimum income
What are you talking about? Universal minimum income is a socialist proposal. It has been for a fairly long time. I suppose they'll be glad to see the libertarians also agree that it could work, but it's not like Hayek alone invented it.