I think I found the culprit.
But, before I get to that, thanks for the tips you gave. The sampling rates between the project and the audio file did not match. So, I corrected that, tested and the problem was still there. Then I tested it on a another computer and the problem was still there. Curiously, though the clicking started about 30 seconds into the video and then vanished about 20 seconds later.
So, I decided to to try out a version where I did not include the FCPX Compressor as part of the audio processing when it exported to YouTube. Now, the clicking is gone from YouTube playback. It appears that the FCPX Compressor was somehow introducing the clicking in playback from YouTube, but not when playing the file in FCPX.
Thanks again for your tips. Your best practices suggestions I will be sure to make use of.
If you have any further thoughts on this Compressor issue, please let me know.