Any concerns with WHD when moving the database or updating the SQL Server and/or Server OS?

Should be fine, as long as on the new Server the connection user has been setup correctly as it will be dba of the SQL Database.

I'm using PostgreSQL and Linux but I recently moved mine to a whole new system (different and newer distro) with no issues. Just shutdown, moved the DB, reconnect on startup and everything came back up just fine.

Closing the loop on this. The migration was very smooth from the application's perspective. Where we ran into minor hiccups is likely common to all database/application migrations. We use WHD slightly off-label and to assist in that, and our custom reporting, we have an agent that reads/writes/executes on various tables, functions, procedures, and scripts. Not all the permissions were correctly set before-hand resulting in a bit of a rush on our side as once the application was turned back on we had tickets flowing but some of the automated tasks (external to WHD) weren't yet in place.   
All things considered, it was straight-forward.