I am also having this issue. I can enroll iPad mini devices, iPhone 4 devices and iphone 5 devices but am unable to enroll any of our iPhone 5s devices.
Switching off private browsing makes no difference for me. I have tried with 4 different iPhone 5s' and like I have already said, have no issue enrolling other devices.
I am enrolling by emailing trust & enrollment profiles to the devices, installing the trust profile (works) then installing the enrollment profile, it says complete on the phones and the phones show up in profile manager devices but the enrollment task hangs in active task in status "in progress".
I even left the active task for over a week and nothing happened.