This is a completely elementary approach that seems to work:
If $(x,x)$ is a point in the diagonal and $(x_n)$ and $(x_n')$ are "essentially different" sequences(*) that converge to $x$, then clearly any compactification of $S_2 \times S_2 \setminus \Delta$ has to contain $(x,x)$. And since $S_2 \times S_2$ is a compactification of $S_2 \times S_2 \setminus \Delta$ and it contains all the points it absolutely has to to be a compactification, it must be the smallest one.
As to whether there are larger ones... probably, but it's fairly obvious that $S_2 \times S_2$ is the Stone–Čech compactification of $S_2 \times S_2 \setminus \Delta$, so that puts some pretty clear limits on what other interesting compactifications you can have.
(*): More formally we could write $x(\mathbb{N}) \cap x'(\mathbb{N}) = \emptyset$.