I have a file; http://www.mi5.com/evidence/video/blair04a.mp4
which plays in QT 7.7.6 (2014).
I also have a file; http://www.mi5.com/evidence/video/c4nws02a.mp4
which which does not play in QT 7.7.6 (2014)., but does play in QT 7.7.1 (2011).
As previously noted almost 10 months ago, the "c4nws02a.mp4" is corrupted—i.e., the header description of the data does not match the data it found in the file. You can either fix the file by re-encoding the file using an app that ignores the corruption and writes header descriptors matching the data (i.e., without corruption) or you can use a player that ignores the corruption like VLC. I prefer the option of "fixing" the file so that it is then compatible with all MP4 compatible media players.
Both files were encoded in a previous version of QT.
Anything from a momentary power surge or brownout to a failing HDD can cause corruption at any time. Either learn to live it or use a UPS RAID storage to protect your data.
Clearly this is a bug in QT 7.7.6.
Not so! This is a "security issue." Apple constantly updates software security for your protection. It is obvious that a security update was added sometime between the two players your reference and that the added security layer assumes the header corruption to be a security threat to your updated system and, thus, refuses to load the file.
Should I inform Apple of the bug in QT 7.7.6? How would I do so?
Feel free to send feedback to Apple but as soon as they duplicate your workflow and see the modal message regarding the "sample" discrepancy, they will likely ignore your feedback since the security update is acting as it programmed to in the event of a possible security threat. As to how to provide feedback, use the product feedback links previously provided above on 18 July 2014 (which are still valid) to select a specific Apple product for which you wish to send feedback. In the case of QuickTime, you can go directly to its form using the following link: http://www.apple.com/feedback/quicktime.html
Alternatively, should I perhaps re-encode some of my video files so they work with QT 7.7.6 ?
As previously indicated, my recommendation is to fix the corruption problem so that the file will play in any MP4 media player capable of playing JVT standard file encodes—including all versions of QT 7 and QT X, as well as, third-party players and devices. Since I normally delete uploaded sample content after a month or two, I have re-converted and re-posted a "fixed" copy of the problem file which plays correctly in all of my players and on all of my mobile devices. The only change I made was to re-encode the audio as mono. URL: http://downloads.walker4.me/Temporary_files/c4nws02a-HandBrake.mp4