O seu quarto está um balaio de gato! = Your room is a mess!

A reunião foi um balaio de gato! = The meeting was a circus! (full of disorder, nobody understood each other)

As far as I know, "balaio" is just a large basket made of straw or wicker (similar to a hamper). It's used to carry and store several kinds of objects. And may also be used as bed for pets.

As for "balaio quadrado," I guess it just means square basket!