What is the best way to losslessly compress MJPEG to MP4?

I have a few digital camera movies recorded in MJPEG that are taking a few GB of storage space. I would like to better compress them, but losing as little detail as possible, or none.

Since MJPEG doesn't compress across frames, I think that any modern movie codec would do a better job, wouldn't it?

Would the following ffmpeg command do what I expect?

ffmpeg -i input -c:v libx264 -preset veryslow -qp 0 output.mp4

Your command will do it but the chief draw of x264 is the ability to provide high compression while maintaining subjective visual transparency. x264 will be of modest use if you keep to strict lossless mode. I suggest trying the following command:

ffmpeg -i input -c:v libx264 -preset veryslow -crf 18 output.mp4

If you don't like the quality, lower the CRF value.