@Brad Roudebush Perhaps there is a third-party addon that is able to change the date when an issue's due-date is updated (I dont't know one, sorry).
Otherwise you might add a new feature request to the plugin's issue tracker here, optionally restricting due-dates to business-dates: https://bitbucket.org/hascode/jira-quick-subtasks/issues?status=new&status=open
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