QuickTime is just a wrapper, not a codec and it sounds like the computers you are trying to play the MOV on don't have the proper codec. When you open the file in QT hit command+I and where it says "Format" it will tell you the name of the codec.
You can think of a wrapper as kind of link an envelope and the codec as the language used to write the letter contained inside the envelope. If someone sends me a letter I know how to open the envelope but if the message is written in a language I don't know I won't be able to read it. That's basically what's happening here.