IANAL but from a quick review of some online authorities, maybe it'd apply. It might depend on exactly how the relevant agencies are classified – are they a health clinic themselves? The restrictions might be waiveable administratively for a popular cause, or violations might have no consequences when happening between chummy government agencies. (Who'll prosecute, and what effective penalties/remedies could be applied, if a governor/president commands coordination between two agencies under their control?) And given the trends in broad warrants, I'm not sure a future court wouldn't offer a warrant to sequence-all-the-spots and search for hits with some heinous criminals' samples.
HIPPA expires 50 years after a person dies, perhaps in the future these cards will be sequenced and crimes with DNA can be narrowed down to descendants.
Good point! This mass, automatic screening began in the late 1960s, so for any states that have retained most/all spots since then, millions of samples would become available for sequencing, despite any possible HIPPA limitations… right about now.