The quality of being somewhat like a square, but not exactly.
It's just a way to talk about things that are a little bit like a square.
Examples:
The squarishnesses of the table make it fit well in the corner.
The squarishnesses of the building give it a modern look.
I like the squarishnesses of the picture frame.
The squarishnesses of the cake made it look like a present.
The squarishnesses of the TV made it stand out in the room.
The squarishnesses of the box made it hard to fit the round ball inside.