As Franklinstein writes, check that the IIcx is complete, with all parts and cables correctly seated. Check that the ROM jumper W1 is present (near grid reference A8 on the MLB), unless you have a separate ROM SIMM card. The autorestart switch behind the Mac should not be engaged. Use a screwdriver and a 90° rotation to let it disengage.
The IIcx, IIci and Quadra 700 share a power supply unit, made either by Astec or Delta. The TRKL (5V) supply, on which soft-power startup depends, goes bad (most often on Astec units), so that the Mac will not respond to the keyboard power switch, or sits, clicks and looks stupid. Trawl these forums for the many references to this failure, workarounds and cures. This PSU has also a less-frequent oddity, from (probably) the same cause, in which either soft-power shutdown or Special/Shut Down stops at the 'It is now safe to switch off your Macintosh', as if the Mac were a passive power-control machine (like a Classic AIO). All you can do then is pull the mains plug from the Mac (and replace it after the Mac has shut down).