Will Intensity Shuttle for Thunderbolt work on USB 3.0 port?

By using a USB-C to USB 3.0 cable.

No. Thunderbolt isn't USB

Is the USBc port a Thunderbolt only port, a USB3.0 only port, or is it a combined USB3 / Thunderbolt3 port?
(As found in Intel NUC's)

The first should work, the second won't, and the last should switch modes inside the port and then work as normal.

EDIT: The port on the PC is the USBc shaped one right? If it's the port on the Shuttle that's USBc/Thunderbolt3, and USB3 on the PC, no chance.

USB-C and Thunderbold 3 use different protocols A USB-C device can talk to a USB-C interface/device through a USB-C cable, A Thunderbolt 3 device can talk to a Thunderbolt 3 interface/device through a Thunderbolt 3 cable. allthough the connector may look the same, the cables are different.
only a good interface converter can make that change, but keep in mind it's only as fast as the slowest one in "the chain".