Yes, they can do that. I guarantee that you are subject to probation terms in addition to the specific ones laid out by the court. Specific MCL 771.3 gives statutory conditions for DUI that include providing of such details. Your "embarrassment" is not going to mean anything to the probation department or the court. You can certainly protest to the judge who issued the probation order, but I don't think you are getting anywhere.

The Probation department is not a covered entity, HIPAA doesn't apply to them (nor would it seem that HIPAA grants any relief in this case even if the department were covered).

What do you mean "use in court?" Why are you going to court? As long as you comply with your probation conditions you'll never see a court room again. I guarantee if you don't comply, you will.