Generally Fe appreciates a little prying from what I've seen. If they really aren't comfortable talking about it, they will either say so or they will change the subject. NFJ Fe users at least often need to know that you actually are interested before they'll go ahead and share anything. This goes for stories, opinions, advice and so on. Ask questions, starting with surfacey stuff. If the NFJ mentions something deeper, that's an invitation they are throwing out there for you to ask about it. If you don't, they'll continue on (although they may note for future occasions that either you don't feel comfortable going there, you aren't interested, or that they can't share more with you). If you do ask, they'll usually tell you.
For me, I need someone to talk to in order to clarify, untangle and sort out my own thoughts. You are the catalyst for me to do this. By asking questions, you help spark my own thinking. It's kind of like House uses his team to find the answers. It's not that they are giving him the answers but something they say triggers something else, or in ruling out their diagnoses, he figures out what else he hasn't tried. You perform a very valuable function in doing this.
Secondly, before I can get to this stage, I need someone to help me bleed off the excess emotion I feel by allowing me to vent about what the problem is. I will probably paint things even blacker and more dismal than they actually are once I've unburdened myself. (Usually when I'm done I try to thank the other person and acknowledge that things really aren't that bad). It's the emotional equivalent of allowing me to lay my burdens by the side of the road when I feel like I can't go on anymore. Cutting off the process or seeming disinterested is kind of like adding a couple more things onto the load. During that time, don't share a lot of your own personal experiences. Just keep listening and asking questions and don't try to tell me things will be alright. I want you to see just how not alright they are. One I'm done, then I will probably start asking about you, invite your advice, or be okay with you sharing something.
I think most INFJs are very open to talking about themselves, but may limit talking about their problems to talking about them once they feel that they have a handle on things or once they are in the past. Before that, we just need your sympathy (no devil's advocating then!) and maybe a hug or something. I will disclose things to people only if I have a very good idea of what their reaction might be and if the lesser things I've disclosed have been received openly and in a way that doesn't feel rejecting. I don't want someone to carry me through a painful experience. Seeing that I'm feeling emotional pain probably means more to me than you trying to help me through it. Some small acknowledgement (like cough candies if I'm sick or something that says you care etc) and frequent checking in means worlds for both physical and emotional pain. One of the things that shuts me down from sharing anything is if someone does something that embarrasses me or makes me feel stupid for liking it or caring about it. If you feel those things, don't let them register on your face (we're kind of hypersensitive to your opinion, especially if you are important to us) and just ask more questions about it to be sure you understand why we like it or why it matters.