I would absolutely not want my DNA on file. It seems like every month someone has hacked yet another "totally secure" database, and in a country where health insurance is a private, for-profit enterprise, that information is far too valuable to place at risk. If I am carrying a marker for cancer or Huntington's disease or something like that, I don't want that data where anybody can get at it.
I can also see a big, data-savvy employer (Google? Microsoft?) using such a database to screen applicants for things like an alcoholism gene; it wouldn't matter if you were a teetotaller, you'd still be blackballed for the job.
I know of someone who was turned down for a mortgage in the 1980's at a small town bank because somehow the bank found out that the individual had gone in for an HIV test. Although the test was negative, the bank made the assumption that the person must have been engaging in risky behaviors, and might die before the mortgage was paid off.