Circuit is broken when switch is screwed into box, but not outside of box
The green wires are the mystery here.  One thing is clear, they are not correctly installed equipment grounding conductors (ground wires).  

If this is conduit as you say, it's possible that it's just a color coding problem, which is an issue because it is confusing and potentially dangerous when people work on the wiring, but not necessarily an electrical problem.  The solution would be to identify where green wire has been used incorrectly and remove it or replace it with a suitable color.  

However, is it possible what you have here is not conduit but armored cable with an insulated ground?  It sounds like you have armored cable where someone decided to use the armor as the ground (not good) and use the green wires for power (also no good).  If that's the case, someone has to sort out how this has been screwed up and correct it, and it could be a big headache.  It sounds like someone partly corrected it - they capped the green wires in this box - but they didn't fix everything.  

Apparently one of the green wires is hot;  when you connect the green wires to the ground terminal on the switch, both green wires are hot, and the switch strap is hot - you're lucky you didn't accidentally touch it when you had the box open and live, you'd have been zapped.  

When you close the box, the screw clamps the switch strap to the box, which is grounded by the conduit or armor;  when you restore power, you have a ground fault and the breaker trips.  

If this is armored cable rather than conduit, it's likely that the cable armor is not a reliable ground path, and you're lucky that the breaker tripped, rather than heating up the armor and potentially causing a fire.  

If you cap the green wires and install a dimmer, everything should work as it did, but you still have an issue, possibly a serious issue.  

Ultimately I think you're going to want have an electrician take a close look at the wiring and the grounding and sort this out.  If it's conduit it may be a simple matter to trace the wires and remove or replace any green wire that ought to be a different color.  If it's armored cable, it could be quite a bit more work, and you may find that other shortcuts have been taken.