Hi JCole,
there were some issues with Logitech and their wireless hardware but it seems this is something .
I am using T400 mouse and K360 keyboard with unifying receiver, and am facing quite serious (and similar) issues with NUC crashing & rebooting (posted also in this forum), but there is no evidence (yet?) that the unifying receiver is the cause of the issue.
Did you try to use USB extension cable and plug in the unifying receiver at the other end of the cable to move it away from the NUC chassis? Upon seeing your post I will try the same, will post back the results.
Regards,
/HarveyV