The most important factor is going to be the competition for EMS jobs in your area. If there are 20 applications for every opening, your application isn't going to make the first cut. When hiring, the first look through of a stack of applications is to weed out those you believe are a waste of your time. This gets the pile of applications into a more manageable size. The application that is poorly written, mis-spelled, bad handwriting, sketchy job history, criminal history etc is not going to make the cut in a competitive market.
On the other hand, if there are only a few applicants, you are more likely to get the opportunity to explain circumstances of the event.
Vent is right though about the 'expunged record'. Juvenile records are generally 'sealed' at 18, but sealed record can be opened if you have the right key. Often with a deferred prosecution, done in a lot of first time DUIs, after your probation period has finished, there is another form you must fill out to have the record expunged. It is not always done automatically. In the case of deferred prosecution, there is even sometimes a narrow window of opportunity in which that paperwork has to be filed.
Its definitely worth checking out with an attorney. If you cannot afford and attorney, one will be provided.... No seriously... most areas have a pro bono legal agency for providing free or low cost legal services, generally based on a sliding scale per your income.
Its definitely worth checking out. A clean record is always going to be better than a tarnished one. If you can make it go away, do so.