If you like Media Player Classic, why not set it to be the default player for this type of file?
Right click on the mp4 file, then click Properties. You should get a pop up window and under the General tab, you will see a line that says Opens with, and a button that says Change. You can set the mp4 file to open with any media player this way.
I would suggest a slightly more advanced media player, like VLC or Media Player Classic Home Cinema. Both are free to use.