It appears that you had a conviction on your DUI. This is not a minor traffic violation (a first time DUI is a class A misdemeanor, which you could have been put in jail for up to 364 days). If you had a supervision, you weren't convicted of a crime.
Can they do this? Yes, they can. If the record came back as a conviction, your failure to disclose it can be the basis for termination (if you've gotten the job already) or denial of your application.
If you need to find out if you were convicted or given supervision, you can get a copy of your court purposes driving abstract from any secretary of state's office (last I checked, they cost around $12 or $13). If you were convicted, it will show there.
DUI DUI as a criminal offense DUI charges DUI and criminal records DUI and employment consequences Criminal defense Criminal charges Misdemeanor crime Crimes against society Defenses for criminal charges Criminal court Criminal conviction Criminal record Employment Termination of employment