As someone with pretty nasty social anxiety, I have some insight on this subject.
There are many reasons why he might not be very talkative. It could be social anxiety (clinical or just shyness) or any number of other reasons. One could even be asperger's syndrome. The reasons really don't matter.
First I think you need to accept him for being the way he is. The only thing that'll work (in my experience both as one with issues and dealing with people who do) is to make them more comfortable and be accepting. You've probably tried building a rapport so keep working on that but don't push too hard.
You need to make him feel "safe". I know that'll seem odd on in a business context but it's true. I would make sure that these one-on-ones are positives always. You may need to chastise him at times, but find a way to do so in a different context than these one-on-ones. That way when you ask him for one he won't internally panic. I would suggest scheduling them regularly so they become routine and always at a certain time. Even call them something like "our monday meeting" so they feel routine as well.
As I've said, I've got social anxiety. I've also been laid off or terminated a few times. It happens to all of us. But for me, I can't help but go into a near panic any time my boss says "can you come to my office for a minute?" even though I'm valued, my work is good and my position is budgeted through 2017. I still panic. Every. Time. I say this because you don't know what history your guy has one on ones could cause discomfort because of history or simply because he may just feel vulnerable.
I would also trying having one that is a "get to know you better" sort of thing that isn't even mostly job related. Open up about yourself. Maybe he'll open up.
Ask his opinion on issues. Ask his advice on process where appropriate.
Be careful of open ended questions. When those of us who are uncomfortable get an open ended question, we tend to answer it as shortly as possible.
Find what he likes and what he's passionate about. None of this takes much time but it's a process.
In short, you need to earn his trust.
When it's all said and done, it may not work. He may just not be the kind of person responds well. Maybe emails or instant messages will be a good option if that fails.