I was in the exact same boat, sort of. I broke a policy that was a conduct issue and was fired for a conduct reason which looks horrible if they called that previous employer. They can't say what I did but can say I was fired for a conduct reason. Thankfully it is not something that would show on a background check.
If you are saying it will show on a background check then it sounds like something illegal. I would say honesty is the best policy here. If they are going to find it on the background check anyway then just come out with it in the interview and if it is a criminal record, don't check no on the application. They are going to find it anyway. Explain what happened the best you can and then list all the steps you took to better yourself. Show that you are taking full responsibilty for it and are already in the mist of bettering yourself. Therapy, went to a drug rehab, etc. If it will show on a background check and you stay quiet about it, they will see it anyway but without your point of view and all the bettering yourself steps you took. Remember, in an interview, you are selling yourself to them, basically. This is no exception.
If you don't know if it is on your nackground check, you can find out.
What Records Will Show Up On My Background Check? | PreCheck and you can do a background check on yourself for a few bucks. My favorite one I use (yes, I background check people) is intelius.com. You can find any public record on someone and any criminal record that is public information.
If you have not taken steps to better yourself, then do so. If it is drugs/medication then go to a dual diagnosis drug rehab which helps both drug addiction and mental health issues. Therapy, CBT & DBT classes. Learn better coping skills. You cannot expect to jump back into the same career and not do it again if you have not taken steps necessary to be sucessful and cope better next time. It is not "I learned my lesson" thing if it will show on a background check. And if it was poor coping skills then the least you need to do is learn, really learn, how to better cope when things get really hard again.
When I lost my job I had a savings and cashed out my 401K which gave me a huge chunk of money. I spent a few weeks sleeping. I was overwhelmed and my body shut down. Try to prevent this. Overwhelming yourself doesn't help. Try to take it slower, if possible. Finding a job should be a full time job but personally, because of already being overwhelmed, I couldn't do that. Do apply for jobs but take it as you can. If you stay in "OH MY GOD! I NEED MONEY SO ME AND MY SON DON'T END UP HOMELESS" mode, that will just cause your body to crash or at least make your mind spin too fast and in the end you likely be less productive then if you took it slow and steady. I thought about it as the guy that speeds past you and swerves in and out of lanes cutting people off on the road in traffic just to end up a few cars behind you at a red light. Personally, I had the money to take it slow. If you don't then try to be at least organized. That helps my brain to not spin so much. List jobs that you see then apply for them crossing off each one. List out each steps you need to take and then do them crossing them off as you go. This also helps me to see that I am indeed making progress.
Also, on Career Builer, when you apply for one job, it finds 50 other jobs like it and you have an "apply to all" button and with one button your resume goes out to them all. I did that twice so in reality I applied to about 120 jobs rather then 20. Try all of the job sites. Also, branch out. What else can you do besides nursing? Any skills? Mike Rowe, of Dirty Jobs, recently said in an interview that everyone needs to have at least one skill (a "dirty job" is a skill) so that if something happens with your career, you always have something to fall back on and it moves with you. Also things like customer service, call center, receptionist, many ways to work at home and make extra income. Don't limit yourself. Money is money and you can always look for a nursing job while working in another field.
Lastly, you do need to forgive yourself and learn that we sometimes fall back on bad coping skills. It doesn't make you a failure. It makes you human. Take responsibilty for what you did but also challenge the "I'm a failure" self talk and replace it with "I fell back into old/bad coping skills. I have learned from that and really did my best to learn new and better coping skills and I will do better next time" or something similar. Keep up on self care and going through DBT right now will also help.