Much of this internet privacy backlash is aimed specifically at Google and Facebook (and of course the NSA, but let's assume that they don't have any problems getting to your medical data anyway).
But then even if they weren't anonymized, I think Google would be one of the company I would trust the most for that. They know what they're doing in terms of security and of all the companies that know a lot about me, they're the one who consistently delivered great value from the data they have. I'm thinking of things like search, Google Now and the maps interface for my location history. Plus I'm happy that the monetization of my data serves to fund great projects like this one. It didn't have to be so...
The only way you can properly anonymize such data is by reducing it to aggregates but then most of the utility is lost.
Taking the combination of google search data and medical data it would not at all be hard to tie some of the searches (and thus identified users) to medical records. Especially not if the diseases / symptoms are rare.
Health data is very far from this idealized anonimization-friendly case. It is some of the most specific data avalible about you, and it can easily be abused to disadvantage you in ways you can do nothing about (insurance, job prospects, social standing, even as far as getting your life threatened by people who disagree with things like sexuality and abortion)
Im sure Google appreciates your faith in them. But you should carefully consider the consequnces. Once the information is out there, there is no getting it back.
[1] Although Google's direction is arguably more stable than is usual for publicly traded companies, given the particular share structure designed to keep control in the hands of the founders.
[2] Mostly, being in generally good health and living in a country where basic health insurance is guaranteed and good enough, with premiums that don't depend on your health.