I had this same issue. However, when I checked the newly installed service (recreated from the update of QB) actually was crashing and failing to start. When I changed the service off from QB user to local system the service started properly. Changed it to automatic from manual and tested it on a test account on my Terminal Server. It is no longer asking for admin rights to open Quickbooks.

Make sure to change the service to manual. It could still be asking to start the service when QB is opened if it is set to manual. And of course a standard user doesn't have rights to access services.msc to start the service.