I know about this other thread about the same problem: NUC6CAYH losing ethernet connection but I thought I could share my own research about the problem.
I have installed BIOS 0038 and Ubuntu Server 17.04 on my NUC6CAYH. I also have reserved the IP address for it in my local network based on its MAC address. This exact setup works for my other local server which is an old Mini ITX Intel board with Atom processor.
The LAN card complete disappearing can be reproduced like this:
1. Set `Wake on Power Loss` in the Power options tab in the BIOS
2. Save the BIOS and wait for it to restart and boot up, after that hold the Power button for 5 seconds
3. The NUC briefly powers off and then it starts again, this time with missing LAN card - even the small LEDs at the back of the LAN card are turned off
4. Even unplugging the cable wasn't working for me, so under `Onboard devices` I disable/enable (depending on the previous state) all other devices except the LAN card
5. After Save in the BIOS, the NUC restarts itself and the LAN card is back online
So at that point I decided to not use the `Wake on Power Loss` option, but then I stumbled upon the `no attached monitor` issue:
If I have a monitor plugged in the HDMI port the NUC boots up and its accessible through the LAN card. The monitor itself is being used on another input source because I'm accessing the NUC through SSH. I left the NUC running for 3 days and I was issuing a `ping` command from time to time just to see if it is still online. This test passed without an issue.
Next I tried several times running the NUC without a monitor. I unplugged the monitor and restarted the NUC just to get it a clean start for the test. The NUC successfully boots up and I'm able to access it through SSH and test it with the `ping` command.
However after several hours (can't tell exactly how) the NUC become unreachable. At this point I can't tell exactly what is going on because the NUC looks like it's working and the LAN card's LED are on, and the one even blinks from time to time. If I plug the HDMI cable back in the monitor won't detect the input signal until I restart the NUC, so I can't tell what the state of the NUC is.