Why does the X rotation on my shape key driver go negative when I'm rotating on the Z axis?

Your confusion derives by two different axis orientations: you did rotate the whole armature (an object which contains only one bone, but they could be more), while the expected result should be achieved rotating only one bone (in pose mode, not in object mode).

In my picture I turned on axis rapresentations, the big ones are referred to the object (the whole armature, that should not be rotated); the little ones are thoose of the single bone. As the two axis system are not aligned, you can see that a rotation on the Z axis of the whole object while it's tilted even on X and Y axis causes a negative rotation on the X axis of the bone.

I also find this system a little confusing, but this is how it works, start rotating bones in pose mode and keep bones axis turned on in viewport display properties to get used to it; drivers are a great resource once you master them!