So this is difficult to get past, for an employer, and that's really the right response. I would prefer that doctors, lawyers, accountants, and other professionals be held to a high ethical standard, and I hold myself to that standard.
Unfortunately for you, you've given evidence that, at least at one point in the past, you have not held yourself to that standard. Your description of the incident suggests that "no harm was done", this is not a mitigating factor. If you break into my house and just look around, this is as much of a violation as if you break in and take my stuff. The response might be different, but the ethical breach is the same.
Fortunately, it is possible to recover, I think. I would suggest that for the foreseeable future, you must be the first person to bring this up. Don't let a potential employer or school be the one to discover this, and don't make them ask you about it. And when describing the incident, don't try to downplay it. "nobody was hurt" is the wrong thing to say. Describe exactly what you did: you defeated school security measures to access information that you were not authorized to access. Let the person you're talking about make the judgements.
One interesting possibility occurs to me, if you decide that medicine is closed to you (it might not be, you should talk to an academic or professional advisor about this). I suspect that there are not a lot of good programmers with significant domain knowledge in medicine. You might be that person. This could be worth pursuing.
Yes, I do realize the degree of trust that is required by all professions including in computers especially. And sadly, I should have thought about this before committing to my mistake. And yes, the "truth policy" is always the best policy. I don't intend to downplay my actions. I have thought about the damaged I have caused. I could have accessed to a lot of sensitive information as well as sadly risked the IT security manager his job, career, and possibly his family. I hope when the time of judgement comes, they will be able to accept me for what I have done and see that I have learned from my foolish and naive self.