The user accounts were using Forms based Authentication.
Users are not given rights to edit or close the Quest webpart.
Forms Based Authentication users not having the above permissions still get the menu to “Close”, “Modify Shared Webpart” on the drop down menu of the Quest webpart.
The SharePoint administrators want to remove this option.
Title
How to Remove the Forms Based Authentication Users Ability to Close the Quest Web PartsDescription
Resolution
The "Close" and "Modify Shared Webpart" menu options is not managed by Quest Development Studio for SharePoint and this may be an issue with Forms Based Authentication.
The normal Active Directory users not having the required permissions are not able to see the "Close" and "Modify Shared Webpart" options.
The only workaround is to put the Chrome to "none" for the configuration of the webpart for Forms based Authentication users not to see the two drop down menus, using SharePoint Designer (SPD)using SharePoint Designer (SPD).