How can I make particles move like a square in Unity?

Create a particle system as a child of the sprite. Set it to create particles in world space, so spawned particles don't move when the system itself moves. That creates a "trail" effect like you see here.

Then create a looping timeline to create an animation to move the position of the particle effect around the sprite in any paths you want. A timeline can also be used to control any other settings of particle effects over time, which allows you to sequence some really neat choreographies for them.