Yeah, sure can do. I've got a lot of video handling requests so, depending on the very specifics of what you want, you may be able to use one of the two codes I ended up refining:
It'd be easy enough to mix those two up together.
What you won't find there is that minimalistic video playback control, which would be very difficult to do. Even having any video playback control and such a fullscreen video might be currently impossible, unless we'd host the video somewhere proprietary. But squarespace can't host videos, so it will depend if that's your case or not (I'm just assuming you're probably using squarespace because 100% of my clients so far were there).