Your emplyer can share the personal information about you with whoever needs to know in order for them to run their business, including HR personnel.
As for the HR person sharing this information, the same rule would go for them.
You may be able to argue that you told your employer the bare miniumum required to explain your absence, and they could have done the same, while keeping your DUI private, but this is arguable.
The practical problem of turning this into a lawsuit, or even a claim for a lawsuit, is that it magnifies the disclosure (besides costing you legal fees). What's being said about your DUI trial is true (presumably), so this would be a privacy claim and not a defamation claim. You'd have to repeat these facts, which you wish no one knew in the first place, and get your co-workers who had this disclosed to them become involved as witnesses, and try to quantify the damage to your reputation and your embarrassment at your co-workers finding this out. Even just sending a cease and desist letter warning your employer and anyone who knows about this not to spread it around might just result in more gossip.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.