Proper medical care should not be subject to an insurance which the insurance companies could refuse to give you due to prior illness. What if someone grows up in a poor family without medical insurance and is diagnosed with a heart defect at a young age which may or may not manifest itself at an older age. If it does happen, they're practically screwed, even if the defect could be treated with proper medical care. The insurance companies could easily say that this is a prior condition that was diagnosed before the insurance was taken out.
We can argue about socialised medicine, which works pretty well in Europe, warts and all, but it baffles me that such a large portion of Americans are highly enthusiastic about a system where a large for-profit corporation with a huge profit motive can decide to screw you over, due to technicalities in your insurance contract, or because you got a test done when you where 13.
The rest of Larry Page's arguments seem nonsense to me. We, as a society, should be less judgemental and prickly or private about a lot of illnesses, but as an individual, you have to live in a society where people will judge you for your medical history. We are not ready for full disclosure and probably never will be.