A *square* in $\mathbb F_2^n\oplus\mathbb F_2^n$ is a quadruple of the form
$$ (u,v)+\{(0,0),(0,d),(d,0),(d,d)\},\quad u,v,d\in\mathbb F_2^n,\ d\ne 0. $$
A set $A\subset\mathbb F_2^n\oplus\mathbb F_2^n$ is *square-free* if it does not contain a square.

What is the maximum size of a square-free set in $\mathbb F_2^n\oplus\mathbb F_2^n$ (asymptotically, for $n$ growing)?

Equivalently, considering the complements:

What is the minimum size of a set in $\mathbb F_2^n\oplus\mathbb F_2^n$, blocking every square?

(Here "blocking" means "intersecting non-trivially".)

It may be worth noting that there do exist small-degree polynomials in $\mathbb F_2[x_1,\dotsc,y_n]$ vanishing on at least one corner of every square; for instance, assuming for simplicity that $n$ is even, $$ (x_1+y_2)(x_3+y_4)\dotsb(x_{n-1}+y_n) $$ (thanks to Peter Pach for this example).