I’m sure this is very worrying. It’s painful to watch someone we love be in pain. Your boyfriend has been through a lot and it sounds like it’s not over yet with both parents struggling. I’m sorry he won’t take advantage of good help that is available. There is no shame in talking to a therapist. However you may care for him, you don’t have the training or experience to give him the help he may need to cope.
On the other hand, you do have something no therapist has: You care about him personally and you can have good times with him. I suggest you not push him to talk about things he doesn’t want to share. Instead, focus on positive things. Get together with friends. Do things he likes to do. Do your best to sustain the happy times. Happiness and positive experiences are the best antidote to sadness and negativity.
And, please, I hope you don’t let yourself get caught up in the drama of negativity. Negative drama only feeds negativity. Don’t beg him to share his secrets. Don’t constantly tell him how worried you are. Don’t try to “save” him. He doesn’t need a savior. He needs a friend. When he’s blue, suggest you go do something active, watch a comedy movie, or go hang out with friends doing something fun.
I wish you well.