Any suggestions as to what tool(s) to use to identify what's holding the shut-down back when my external drive is connected Secret-Squirrel?
Process Explorer may help. You can download it from here. It doesn't need installing.
Make sure that your drive is connected then launch Process Explorer (by double-clicking the procexp64 icon for 64-bit Windows). Click its "Find" menu then choose "Find handle or DLL". Type the drive letter for your external drive into the search box (F: for example), click the "Search" button and all processes that have a handle on the drive should be displayed below.
I run Malwarebytes regularly and that hasn't thrown up any problems.
I've never suspected a malware problem.