I just spent some time in a chat session with an Apple support rep. They recommended resetting the network settings. That did not work. Then we deleted the Outlook account and recreated it. That did not work. Then we deleted the account and then reset the network settings and then recreated the account. That did not work.
I have a colleague at work running iOS 7 on his iPhone 4. The flagging is working normally for him. Apple told me to contact my work's IT department but they are scratching their heads.
Strangely, I now see that the problem is confined to flags that initiate from the phone. When I flag something on the computer's Outlook client then the flag makes it to the phone and if I unflag that message the message is then unflagged on the computer. But if I flag an email on the phone to begin with--it just sits on the phone. My IT group says they have no explanation at this point.
If anyone learns anything more helpful, I am all ears.