I was able to replicate the issue on an Intel NUC D34010WYKH. Similar issue as OP only Windows 10 Pro 64bit, reboot and you lose network connectivity. Had to disconnect cable and reboot as mentioned above. Disabling adaptor and enabling adaptor did not work. Removing Ethernet cable did identify a network cable was unplugged, but plugging it in did not resolve the issue.
Same NUC reloaded (different SSD to quickly test back and forth) to Windows 7 Pro 64bit not having the issue. I installed latest Intel drivers for the test.
Of interest, I double checked the device manage on Win 10 Pro, on the Intel Ethernet Connection I218-V drivers, they reverted to Microsoft Drivers. 3/20/2015 (126.96.36.199). Not sure if the Intel installer failed...
Will try to update the drivers to the latest Intel version again.
I was able to resolve the issue by loading the correct Intel Drivers. Not sure why they reverted to a Microsoft driver.
Unfortunately, the driver download page for the D34010WYKH does not have the driver. My adapter is the i218-V. Yours may be different.
One way to ID the adapter:
Right click on START
click on DEVICE MANAGER
click the chevron (>) next to NETWORK ADAPTERS
identify your LAN adapter (as mentioned mine was i218-V), open the device and double check that the driver is signed by Microsoft, not Intel, or that the Intel driver is older than June 2015.
Then go to Intel driver site:
enter your adapter and look for drivers.
Mine was Intel Pro/Lan 20.2 version. Downloaded and installed and resolved the reboot issue.
Hope this helps. Frustrating situation.