Question

Does the script rule create a connection to an SQL database from the Verification Station or the Processing Station?

Answer

While the document is going through the non-interactive stages of the workflow, the Processing Station is responsible for executing the rules (e.g. on the Recognition workflow stage, or after changing the field value with the Script stage). 
The rule execution on the interactive stages (Verification, Exceptions, etc.) depends on the station type: Rich (thick client), Web HTML5 (thin client), or Remote. The Rich and Remote clients execute such rules locally and the Web client - on the Processing Station side.


More information about Rich, Web, and Remote stations differences can be found here:
Differences between stations (sorted by deployment type)