Unwrap. Meaning peeling the surface off of a solid and laying it out flat on the ground.
If your requirement is that there are no duplicated texture coordinates at all, it would mean that you are trying to peel an orange without cutting its skin. Even if you manage to "beam off" the skin in one piece you end up with a hollow shell that you cannot lay ...
As Rob says, you should use the Project From View mode when you unwrap.
Import your image in the the UV Editor. Move your camera in front of your box until you get the view you want. In Edit mode make sure your box is subdivided enough, otherwise the projection will bug. Press U > Project From View.
Move or scale your UV as you want in the UV Editor. ...
There are a few issues.
They are easy to repair, but will take some time.
1) Your original maze contains overlayed geometry.
That's a big problem in 3D modelling.
You will need to clean it up. It's technically very easy, but might take some time.
Clean up overlayed geometry and connect all vertices like this.
2) Keep quads whenever possible.
I cannot ...
≫ I want the uv to represent vertices too, so each vert has a unique uv.
If the word "represent" is to identify which vertex is it, then it is doable.
A cube is almost the same thing to a UV sphere(g-0). You can project from view to make all uv shared by all edges and faces connect to it.
So we get 6 uv values only? No.
Actually, we still have 24 uv ...