Speaking as a person who hates talking about my emotions, and hates telling people the deepest personal stories of my life, the best way to get a person to open up is to shut up and listen.

A person who is afraid to share is afraid because they don't know how you will react, or if you can be trusted. They are afraid if they tell you who they really are, they will be rejected.

You have to show, through time and consistent friend actions, that you are open to what they have to say; that you will not judge or judge negatively the things they think or have to say; that it is safe for them to bring their darlings out into the open, and that you will not smash them.

Don't try to pry out of them what they don't want to talk about. But do be open with your own secrets and vulnerable spots. When you are vulnerable in front of someone else, it makes them feel more safe: you have risked something, you have put your skin in the game, which means you trust them enough to tell them something which could hurt you, and it hasn't, and maybe in return they can begin to do the same.