Replacing GFCI receptacle behaving like the old one
Do you have a voltmeter? You need to check voltage from the black to the neutral and to the ground, also ground to neutral. If the new GFCI is not "clicking" that usually means there is no or incorrect voltage at the location and the GFCI is not resetting. If you really think it is the GFCI then connect it up to another GFCI or standard receptacle that is working and see if it works. 

Also make sure you are not series connecting GFCI's. Make sure there are no other GFCI or AFCI devices ahead of it.

Good luck