Did you enable the headphones in the NU Audio Control Panel? That click that you here on power on tells me that you at the very least have it configured to output to the speakers (they are what makes the click). If you go into the NU Audio Pro software you should see where it says 'playback' on the main window make sure that the icon for the headphones is selected, this determines audio output. Also make sure that you have the volume set correctly for the headphones... After enabling the headphones as a playback device you will need to click on the image of the headphones, this will change the volume slider to say 'headphones' at this point you will be able to adjust the sound levels. What kind of earphones are you using? I would recommend starting out with the volume very low and gradually turn it up so that you dont damage your earphones/ears. 

One more thing, I have the 'Bar-A' and 'Bar-B' RGB zones set to sonic reactive and they light up fine when i set the lighting on them, but they don't light up if i try playing any music.
 
Update:
If i start a Youtube video right at the very begginning i can sometimes get about a quarter of a second of audio to play, but then it cuts out immediately.  The RGB zones do light up for this amount of time when it plays a fraction of a second of audio.  It seems like something software wise is going on, preventing the card from playing.

Oops.. you're right.  Thanks for pointing that out.

I just wanted to make an update to this thread, as I made another attempt recently to get the $300 dollar brick in my computer to function with complete success. Previously, I had only been trying the PCIe 4x slots on my motherboard, as the PCIe 1x slot is inaccessible due to my NH-D15 CPU cooler covering it. So, I figured I'd just try installing the Nu Audio Pro card into the bottom PCIe 16x slot (this should technically downgrade both 16x slots to 8x, but oh well). Much to my surprise at this point, it was finally able to produce sound through my headphones! Albeit, there was an occasional 1/4 second dropout/stutter in the audio every once in a while and anything other than the 16bit 44100hz setting would make it worse.  I even messed with the ASIO buffer size to help mitigate the issue. But, that wasn't able to eliminate it, just minimize it. So, I did some more research on the possible cause which lead me to a thread in this forum that mentioned going into the UEFI and disabling C-States. After doing this the audio dropout issue went away completely and I'm now able to play any bit depth and sample rate without an issue.

Keep in mind that disabling c-states isn't without some consequence. Your CPU will not be entering lower power states and therefore will be using more power and consequently generating a bit more heat and the effects that come from that (CPU longevity, etc.). Some, albeit mostly older CPUs, actually throttle down when c-states are disabled. Disabling c-states is neither a good long-term workaround, nor "fix" for real driver/hardware issues, I'm afraid.

The rear headphone port has never worked on my card.
 
However, the front port works fine, so I use that.