You never said what year you had, so assuming 03-08. If 09 to 10, this may not apply.
There are a number of things that can prevent 12V getting to the clutch. A lot has been discussed above and in
viewtopic.php?f=28&t=37869
There is a clutch relay in the underhood fuse block. See the picture above.
With the car not running pull that relay and use a wire to connect 12V from the battery to the relay socket terminal marked 12V to clutch. If the clutch engages, then you will know that the clutch and the wire going from the fuse block to the clutch are ok. If the clutch doesn't engage, then either the clutch or the wire from the fuse block to the clutch is bad.
To test the wire,disconnect the wire from the clutch, and either 1) use a multimeter to test for connectivity between the wire end and the To clutch terminal of the relay socket or 2) use a wire to put battery + on the To clutch terminal of the relay socket and use a multimeter to look for 12V between the end of the wire and ground. Seems like the probability is low that a problem would be in the wire.
If that all tests ok, then with the ignition ON (not off or acc, and the engine need not be running, as long as the HVAC blower will run), use a multimeter to check for 12V between the clutch relay socket terminal marked 12V from Fuse and ground. If you don't get 12V there, then check or replace the 10A AC fuse in the unpleasant to access under dash fuse block. (I suspect that if that fuse was bad, the AC switch wouldn't light up when pushed)
Now if everything so far checks ok, use kostby's trick from that other thread to put 12V between the two smaller silver contacts on the relay itself, you should feel the relay activate. If it does, then while activated you should be able to use a multimeter to check for connectivity between the two larger contacts on the relay
When the engine is running and you hit the AC button on the dash, it sends a signal to the car computer. The computer should then put out a ground to the AC clutch relay coil, the other end of the relay coil is 12V from the AC fuse. This will engage the clutch relay, and send 12V from the AC fuse to the AC clutch, The AC clutch is an electromagnet, the other end of it gets ground from mechanical connection to the engine block. When the 21V hits the AC clutch, it pulls in the shaft of the AC compressor, where it grabs the otherwise fee spinning compressor pulley. All the electrical stuff above could be fine and the AC still not engage if the car computer decides to not send a ground to the AC clutch relay. it might do that if the refrigerant pressure is out of tolerance, or perhaps for some other reason. Hope2BuyVibeTomorrow above tested his electrical parts, but then had to add refrigerant to get his working again.