I encountered the same problem yesterday and found the solution after seeing this same issue posted on Apple support over multiple dates since September. I followed the advice of one commenter and did a restore procedure of the iphone and that solved the issue. The commenter provided a link to a non-apple web site and the instructions differed only slightly from Apple's instructions here: http://support.apple.com/kb/HT1414. I can't find the original advice or that site right now but here's what I did:
1. ensured that only the usb end of my sync cable was attached to my computer with itunes open.
2. held the home and top buttons down simultaneously on my untethered iPhone to power it off.
3. Keeping my finger on the home phone only, connected the iPhone to the other end of the sync cable that was connectd to my computer (fyi, I use a macbook pro and back up everthng to iCloud)
4. the warning message came up in iTunes about are you sure you want to restore all your media and other data (on the phone) will be erased. You can click cancel or restore; I chose restore. It took a while but all my apps, photos and music returned to my iPhone without the red circles with red squares on some tracks.
Background: I updated to ios 7 back in September with no problems. Yesterday, I imported new digital audio to my itunes library and then went to sync, it appeared that I was way over capacity on my iphone. I tried removing pictures and apps I don't use. That brought memory down, but the sync would not correct the red circle/squre issue of certain tracks in my music on my iPhone. These files played fine in iTunes on the computer. On the Apple thread where I found the advice that I've posted that worked, that person had spoken to an Apple person on the phone who believed that a file must have gotten corrupted on the iPhone and recommended the restore. I think the same happened to me. The new digital audio files which I had imported into iTunes were mp3s which worked fine in iTunes, but must have gotten corrupted during the first sync. I converted to aac and shrunk to 128, but whenever I tried to sync, things looked normal (except for a diagnostic message) but the waiting to copy took way to long and when things finally started to copy, it was so slow that he sync just quit on its own. THEN I noticed that all the music that had the red circle/squares were files that I had originally put into iTunes from CDs - so, music not purchased via iTunes. These were imported quite a while back. Fortunately, they were all backed up to the cloud and restore of the iPhone worked. All that music is back and I don't get the diagnostic message or slow wait and quit problem now.