I ended up fixing this issue!
I found a computer that had not installed the 2011-006 Security Update. I went to System>Library>Quicktime> and copied the file “QuickTimeComponents.component”
I then found this same file on the computer that was giving me problems, deleted it, and placed the old file in the same folder (trying to overwrite the file did not work – I had to completely delete it before replacing it).
All MOVs now play in QuickTime, FCP, Finder Preview, etc. Hope this helps anyone else with a similar problem!