Describe three events in A Beautiful Mind that reflect these signs and symptoms of schizophrenia: hallucinations, paranoia, poor hygiene, irrational behavior, break from reality, and/or violent episodes.

I selected the following scenes from the movie A Beautiful Mind in which John Nash exhibits all of the symptoms listed in your question.
In the first scene I've selected, John Nash is inside his room in the psychiatric facility (which he believes to be run by Soviet agents) and is puncturing his wrist with his own hands. John tells Dr. Rosen that his implant is gone and that he can't find it anymore. This scene shows us John's break from reality, practicing self-harm to look for an implant that didn't exist from the start, not caring for his own hygiene and safety, and exhibiting a severe case of paranoia.
The second scene I've chosen is when John's wife finds out that John is once again working on his "assignment" against the Soviet Union. This is the scene in which she realizes the full extent of John's psychological condition and that he is back to his paranoia. As she runs back inside the house, we see John's break from reality lead him to almost get his baby killed in the bathtub that was filling with water; his wife gets to the baby just in time to save him. Then, we see John committing a series of irrational actions while fighting with Parcher, who, in John's hallucination, is carrying a gun. While trying to protect his family from Parcher, John ends up pushing his wife and son to the ground, further scaring her in the process. After frightening his wife, he runs after her and jumps in front of her car in the middle of the rain, trying to stop her from going away.
In the third scene I chose, we see John during a violent episode. When Dr. Rosen tries to take him to a mental hospital (a request from his wife) in a non-violent way, John punches Rosen in the face while trying to escape. This is because John's hallucinations with Parcher have led him to believe that Dr. Rosen is the leader of the Soviet agents.