I read that Handbrake will do this, so I downloaded the latest version and tried, but Handbrake would not recognize my older .mov files as a valid source.
When I re-made some of the movies from their original Dicom loops, the new .mov files were read and were converted without problem.
HandBrake does not access your system's codec component configuration so many codecs supported by QT on your system may not be compatible with HandBrake while formats not supported by QT may be supported. (I.e., HandBrake supports FFmpeg compatible compression formats.)
Quicktime reads and plays all my .mov files. I do not have QT Pro; must I use that to convert my older .mov files?
Only if you wish to use the QT 7 player app to do your converting. If GarageBand is installed on your system, it can also be used to access QT 7 conversion options. Alternatively, if the files are compatible with QT X playback, then you could use the QT X player to export files as H.264 content in M4V containers and then manually change the extension to MP4 if desired. Most users who don't require layering/compositing features simply download/install the free MPEG Streamclip app to perform QT 7 conversions without purchasing a QT Pro key since this app also uses your system's QT codec component configuration for conversions.
Why are some Quicktime movies recognized by Handbrake but not others?
See explanation above.
(BTW same is true with VLC).
Like HandBrake, VLC employs its own built-in codec components rather that your system's QT codec component configuration and is also why it can play some files natively that QT cannot handle (e.g., BD content in MKV file containers).
Any insights and help would be appreciated. I am using Lion on a Macbook Pro.
You actually have many options available here and will likely have more trouble deciding which is best for your particular needs than finding a particular solution that works. Personally, I keep QT X, QT 7 Pro, HandBreak, MPEG Streamclip, GarageBand, iTunes, and VLC installed to cover all bases and maximize my options when more than one converter can handle the job.