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Actually, Flash and h.264 are two different things.
Flash is, of course, being phased out, but to make use of it, see here
https://www.wikihow.com/Upgrade-Adob...-on-Linux-Mint
Did you choose to install the media codecs during the Mint installation? If you did, then you should have .h264 support. If you didn't, you can install from the menu: click on Sound and Video - Install Multimedia Codecs.