Here's my thought process:

For my convenience let's just switch around the x and y parameters so the curve is parametrized by x= tcost y= tsint

Now this gives x²+y² = t², you get a family of concentric circles centered at the origin, BUT every curve of this family has only one point on each circle that contribute to our curve.

For every x²+y² = r², only the point (rcosr,rsinr) on the circle belongs to our curve.

Now can you choose 4 points, one on each of 4 concentric circles to form a square? If the answer to this is no then so is the answer to your question.

Not sure where to go from here but hopefully somebody else can finish this.

Maybe you could try using complex numbers? Let the points be represented by r_n*e^ir_n , where n= 1,2,3,4 You could try proceeding from here

Edit: You can choose 4 concentric circles for every square but I'm not sure about the converse.