I don't see why not. If it's something that you are passionate about. It may be for your own well being that they are recommending a less stressful job and not because people with depression can't be nurses. You may witness pain and tragedy and be required to work with very little sleep and nurses who end up taking these tragedies and pain home don't do very good and do suffer from depression from their job.
Every state has its own nursing board and its own requirements so you will need to check your state's requirements for specifics. There is a criminal background check before getting a nursing license. Serious crimes would probably preclude you from being an RN. Serious mental illness could be a problem but minor mood or anxiety disorders should not disqualify you from being a nurse. Nursing can be a demanding, thankless, high pressure and high liability profession so the more important thing to ask is if you yourself feel that you are up for it.