In Mail for iOS 7 you can tap on
Archive Mailbox then type the word
flagged in the search box. You can then pull down on the results list which gives you the option of
All Mailboxes, or
Current Mailbox, which is actually pretty useful. Selecting
All Mailboxes should show all flagged messaged globally, and
Current Mailbox only the ones in the Archive that have been flagged. I realize that this is more of a work-around than the way it's done for the normal Inbox, although seems to work relatively well considering there's no actual Flagged button for messages in the Archive.
I've found that the Flagged button doesn't always show all the flagged items in the Inbox anyway. It appears that it has something to do with flagged messages that are stored on iCloud or possibly doesn't search for flagged messages past a certain date. I usually end up typing
flagged in the search and sure enough it finds items that weren't listed when I had initially tapped the Flagged button. After the search completes then older flagged items end up showing when tapping the Flagged button.
*I believe flagged messages are also put at the top of the message list by default when you tap on the Archive Mailbox (at least on mine). If someone has a better solution it would be nice to know, however, in meantime hopefully this helps.