Jitters

Just a few thoughts here, I know you have already played your second game by now, I hope things have improved for you. Just a few things you might be able to work your QB through:
QBs in general place a great deal of pressure on themselves in addition to the expectation that is already on them to not make a mistake - not from you necessarily. The position requires no mistakes for success, and let's face it QBs don't like to look bad. So, when we do, we become a needy blackhole of inwardness and selfishness. A 16 year old kid under that kind of pressure will implode mentally without input.
You could say something like this:

"Son, there isn't anyone in this program who believes in you more than me, but your job is not to score touchdowns, win the ball game, make all the plays, or to be perfect. I want you to be passionate, not perfect. Perfection is a mythical goal, passion changes men, it encourages men, and wins football games. It is your job to have a passion for your teammates and their performance, not your performance - this isn't about you. I need you to Focus on getting 10 yards at a time(first downs), not touchdowns, as that is how we score - we run out of grass doing that and they give us points. Short term objectives reduce pressure, and that is really what we want from you anyway(first downs). You need to realize that the only person on our side of the stadium that doesn't think you can do this is you. I have played my hand, and you are my ace. You wouldn't be in that position if we didn't believe in you. Mistakes are inevitable, losses because of those mistakes are not. Your job is 4-6 seconds of 4-6 yards and a pile of dust! Basically, You run about that far on average and you give the ball away to someone else- you are the UPS delivery guy! - No glory, no glitz, just a basic uniform, a truck, and a job to take care of the package and get it there on time. Finally, this doesn't have to be harder than that, you need to stop listening to yourself and start talking to yourself. Stop listening to your failure, to your anger over missed chances, to your emotions that drive your failure, and start talking to yourself - It's not about how I looked on that last play, it's not about me. These guys need me to lead them and encourage them, not win the game for them. They just need the ball to do their thing, that is my job - Mr. UPS - get it where it needs to be. I am all about 10 yards right now, how far do we have to go, I am Mr. Down and Distance!"

Coach, I know this sounds campy and this isn't self esteem development, this is teaching a 16 year old to develop focus and mental toughness. We tend to romanticize our own ability at 16 to deal with pressure, often overestimating our own mental toughness at their age. We know this isn't physical, but they really just don't know how to think, and they won't admit it to you, they don't want to disappoint you with their weakness.

Hope this helps a little.