Hello @wafehling
ZBrush has many Axis/Grid/Surface snapping features in a wide variety of situations. You’re simply not working with one of those situations. 
Keep in mind the geometry does not exist until the skin is created, at which point the mesh can be modeled and shaped with the broader ZBrush toolset that contains many options for snapping and axis movement.
It sounds as if perhaps you’re trying to use the tool outside of its intended range of use. ZSpheres are intended as a quick and dirty way to block out a rough, primarily organic, low poly basemesh to serve as a platform for further sculpting using simple screen plane movements. They are not intended as a tool for the creation of precise geometry or for the user to have to spend a lot of time carefully positioning ZSpheres to get exactly the tolpolgy they want.
An analog in traditional media would be the quick shaping of a wire armature to hold the clay. There’s no need for the armature to have precise form–it exists purely as a platform to hold the clay and sculpt on.
ZModeler is the tool for the creation of precise low poly geometry with deliberate topology. You can even quickly block out hard surface forms with Gizmo+ masking + Clip/ Slice brush+ frequent Dynamesh + Zremesher for finished topology which will provide more options for constraining movements. ZSpheres are for when you want to quickly create something in the general ballpark of your intended form, and get to work shaping it further with the broader toolset.
All that said, you can constrain the movement of the ZSpheres on the screen plane by disabling the XYZ button in Transform> Modifiers and enabling only the axis you wish them to move along. If you enable symmetry when drawing your ZSpheres and draw them where the symmetry converges at center while holding Shift, you should be able to use this to create more precise movements when Shift-snapping the camera to a specific axis.
The only reason I didn’t mention this right off is because I didn’t want you to mistake this for a recommendation. It is merely possible. My recommendation would be to use a feature that is better suited for the level of control you want.
Please submit any feature requests to ZBrush Support.
Thank you!
Keep in mind the geometry does not exist until the skin is created, at which point the mesh can be modeled and shaped with the broader ZBrush toolset that contains many options for snapping and axis movement.