Making the healthcare data public in an anonymised format would not affect people's jobs, unless there is some property of that data that makes it easily identifiable.
Look back to how we handled HIV/AIDS in the 1980s. Enormous amounts of fear (understandable) and utter disgust-filled loathing (not so much). There were politicians calling for AIDS sufferers to be rounded up and put in quarantine camps but refusing to engage in public health initiatives because they thought that warning people to use condoms and engage in safe sex would "promote" homosexuality.
Up until 2010 in Britain, it was legal for a dentist to refuse to take an HIV+ patient even though the professional association for dentists had told dentists that it was safe to use the same procedures around sterilisation and safety. It also made no sense as it just led to people lying about their HIV status to dentists in order to get access to health services.
I know people who have faced discrimination in work over chronic mental health conditions that they are being treated for. People who have held down other jobs perfectly well, who take less sick leave than people without any health conditions. There is still significant social stigma around mental health conditions: nobody considers you morally suspicious if you have, say, eczema or hay fever, but if you have depression or schizophrenia, some will consider you too "risky" to employ etc.
So, yes, "please make all your medical records public" sounds nice. I've spoken to friends who, thanks to people like Schmidt and others breathlessly promoting this kind of thing at TED talks, think that anyone who wishes to actually exercise their medical privacy to be "irrational". And that medical privacy is a silly idea that we wouldn't choose to have now and that we only have for historical reasons. That the bright sparks of Silicon Valley can't see the importance of medical privacy is really concerning—it shows they've failed to look at the history of discrimination and fear-mongering against people with particular health conditions. Or, worse, it shows they've thought about it and don't give a shit about protecting people from mistreatment by both the state and by other individuals based on their health conditions.
There's something deeper here - Larry Page is thinking in terms of "big data" - "wouldn't it be nice if we could run all kinds of multivariate analyses on everything about everyone?" - but it's fundamentally impossible to do that while preserving anonymity. The same information that might reveal interesting correlations is vulnerable to correlation attacks. Say you wanted to know how nutrition affected immune system response. You'd look at things like height, weight, diet, frequency of minor illness, and rough location (city level, to control illness rate against those around you). To be of any use, the "anonymized" database is going to have to have all those variables correlated per person, which means if I know you're 6'1", eat a lot of bananas, live in Silicon Valley and had a cold last year, I can potentially look you up and find out other things about you.
It's worth remembering that whenever a high-level Googler talks about the social/medical potential of large-scale data analysis like that, that it is inherently hostile to privacy.