Pull the film out of it, set the ASA dial to the highest possible setting, point it straight at a bright light source, and attempt to trip the shutter. Do this with the back off so you can see if it works through the lens. If it doesn't go, the meter cell may be dead or the connection from it to the trap-needle mechanism somehow obstructed or broken.
It's not clear to me if the EE-S has a manual aperture setting for flash use. If it does, turn the dial to set the aperture manually, then try to fire the shutter. This does not rely on the meter cell, so if it works here but not in Auto, you know there's a meter problem. If it doesn't work here, then you probably have a shutter problem.