after a user has surfed to http://myjss.host.com:8443/enroll and started the enrollment process (i.e.; downloads and launches the QuickAdd .pkg installer), does the client's browser need to stay running to complete the enrollment process properly?
The QuickAdd from the /enroll page has a unique enrollment invitation attached to it, to link the assigned user (so when run, and you look that the machine in the JSS, you can see a user of that computer). Other info, such as assigned site, may also be included. A QuickAdd from Recon may not have that user info, or it'll be the same for everyone.