If it's a container:
* Cushioned box: This is a literal description.
* Softbox: Used for photography, it's a light diffuser.
* Beanbag: If it's small and used for seating.
* Pillow case: If it's fabric and used for storage.
* Pouf: A plush, round, footstool-like container.
If it's a room:
* Foam room: This is a literal description.
* Playpen: If it's designed for babies.
* Cubby: A small, enclosed space.
If it's something else:
* Huggable: Emphasizes the soft, squishy texture.
* Plush: A general term for softness.
* Cuddle puddle: If it's a place to relax and snuggle.
To give you a better answer, I need more context! What is the squishy four-walled box used for?