Not sure if I understand fully what you want to do. But I think you are saying that you will build a box, put the touchscreen in it and then mount the whole assembly on the wall?
I think there are a couple of ways you could handle this.
1. Reverse your thinking - instead of having the front of the box open up, I might make it so the back could be removed. Maybe a set of slots dado'd or routed into the mdf so that you could mount the back panel to the wall and then slide the rest of the box onto it.
2. You could follow your first idea. I think you are looking for what are commonly called french hinges. These are the hidden type that are used for the inside of cabinet doors. Put those on one side and then a set of catches on the other and the frame should stay secure.
Digression - just thinking, but is the touchscreen thin enough to mount it inside the wall - so the face is flush with the surface? Not a big job, but a very cool effect!
Just some ideas, hope they help.