Let me get this straight: because you're afraid that the NSA gets the hand on their data you want government regulation?
That's both the government.
When you're giving blood to some entity and you want to safeguard what happens to the result regulation won't get that for you, but independent oversight, sample anonimization at the source (insofar as this is possible with a blood sample, your DNA is your identity), decentralization of test locations (preferably as close to the source as possible), proven sample destruction and no keeping of data post sample destruction would be the way to go. Government oversight pretty much guarantees government access, the opposite of what you wish to achieve.
It will be very hard to create a service like this without a loophole that can be used to sneak such data out.
The best way in which this could be done is by having the tests made so foolproof that they can be done directly in the hospitals without centralized laboratories.
As for the board, you could say something similar about DropBox but I don't see them giving three letter agencies access to your account either without proper process and a court order.
Anything less could sink the company and given the amounts of money at stake here you can bet their competitors would love to do just that.
And who is to say that their competitors are not already acting in a similar manner anyway, they are after all processing about 100x more volume today.