It's a great idea, but before you invest any time in making this happen, I'd suggest addressing the privacy concerns somehow. I know that in the US, the privacy of electronic medical records is governed by a law called the Health Insurance Portability and Accountability Act (HIPAA). It's pretty strict, I think. Could the material be encrypted somehow before going on the blockchain? The issue is that the hospitals and insurance companies, etc., would have to be satisfied that they would not be in any HIPAA danger, otherwise they would get sued. Outside the US, there may be places where you'd have less of an obstacle. Maybe someody else knows more about those?
Here is the agency's summary of this part of the law:
http://www.hhs.gov/hipaa/for-professionals/privacy/
The reason I know this is because I had some records leaked once!!!
HIPAA is the big one, but it doesn't end there. FERPA for college students. And some state laws also. Or just stay away from the freaking US bureaucracy and do it somewhere else. Your idea has merit, but medical records would be the most difficult place to start. I'd start with some other types of records, putting those on the blockchain. Lots of them are public: birth, death, marriage, wills, etc. Try to find a business niche there and then expand it, would be my suggestion.
Well, technically, once your medical record is in your hands, it is up to you to secure it. That's part of the beauty of this solution. The goal would be to put your medical record in your hands securely. Once that is done, it's up to the individual to keep it secure...or not.