Obama's plan is a third option. The health plan will create a government option to make it available to those who can't afford it.
So it'll be sort of like the postal system. If you want to use Fedex or UPS, you can, if you want uber cheap you can go with USPS, but you don't see Fedex or USPS out of business do you?
Responding to comment below: And so many are still without health care! So this will alleviate that. But this will also make it cheaper for large companies like General Motors because they can now switch to the cheaper government plan. It will allow companies like GM to be more globally competitive because european (not all?), japanese auto companies do not have to worry about health care costs.
If the gov plan uses taxpayer money, it doesn't necessarily mean that private insurance will fail. And this is because there will be some who will want the higher quality private insurance. The fact is the richer you are the more you will pay for even modest upgrades. So those who can afford private insurance will still have a reason to go for it. This is evidenced by what TomOfTTB said above who said there are stories of Canadians who fly to the US for health care right? And it's true private insurance will get hit hard by this, but they will probably not all die out. In fact private insurance companies are probably the main driver of the opposition.
So it'll be sort of like the postal system. If you want to use Fedex or UPS, you can, if you want uber cheap you can go with USPS, but you don't see Fedex or USPS out of business do you?