My head is bleeding from where I've been beating it against the wall trying to convert a ripped MKV file (ripped with Handbrake to maintain closed captioning) into a MP4 file that maintains the captions in a hard-burn.
The solution I finally found, freemake.com, is worth sharing. It is a clean and easy piece of software that is full-featured donation-ware. If I were a dog my whole body would be wagging, I'm that happy to have stumbled on it. I installed no fewer than 8 converters trying to find this one!