Phyllis Schlafly responded to this question, but there is something I would have added if I were responding to this question.
As vaccinations for sexually transmitted diseases like HPV--which haven't been well tested--are mandated, I worry that as some students cannot attend public schools with out such vaccinations, citizens might not be able to renew driver's licenses or board planes.
This is only one example of how allowing the federal government to keep and access our health records would make it tempting to "encourage" citizens to get their vaccinations by excluding them from privileges like travel, driving, or something else.