I did everything according to the instructions for the Bottom Interior 1x1 Eyelids but the blinking animation is not working.
I have also turned on "Allow skin transparency: All skins" and enabled the resource pack. My skin file is attached below:

Note: It does work with the Just Expressions mod, but doesn't with Fresh Moves

Problem solved. Turned out I had the Eyeless version installed (It was the latest version released so it installed it automatically)
Make sure to check if you have the correct version installed!!!