I cannot play .mov files (e.g. recorded with an iPhone) in Safari with mediaelement.js 2.10.1.
I get only a black box:

Does mediaelement.js support playing .mov files?
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It does not support it natively but you can modify MediaElementJS by adding support for video/quicktime under the plugin you wish to use.
but because MOV might contain codecs combinations that are not supported across all browsers and plugins this is not recommended.
However, if you know for a fact you are only going to be looking at MOV files recorded using H.264 and on safari you can make the modification keeping that in mind