How to do a screen like this? Answered

My choice would be to 3D print the squares and mount an LED in each one. Light scatter is a big problem with multi LEd displays. You could hand make the same thing in something like plasticard.

You mean the housing with the clear bits for what is displayed?
That clock seems to be an integrated system, meaning all is part of the housing.

Try this:
Place your matrix behind a thin sheet of acrylic or glass.
Cover the other side with tinting foil as used for car and house windows, black limo would be the preferred option.
If you can eliminate stray light than even something with just 50% tint should give you good results.