I will write $β$ for the Stone-Čech compactification. Let $p$ be the canonical map $β(ℕ^2)→β(ℕ)^2$. Let $(u,v) ∈ β(ℕ)^2$. If $u$ or $v$ is in $ℕ$, then $p^{-1}(u,v)$ is a singleton. Otherwise, $p^{-1}(u,v)$ contains at least two points:
Proof. Let $(x_i)_{i∈I}$ and $(y_j)_{j∈J}$ be two nets converging toward $u$ and $v$, respectively. Let $A := \{(i,j)∈I×J\,|\,x_i<y_j\}$ and let $B := \{(i,j)∈I×J\,|\,x_i>y_j\}$. Let $a$ be a limit point of $(x_i,y_j)_{(i,j)∈A}$ and let $b$ be a limit point of $(x_i,y_j)_{(i,j)∈B}$. Then $p(a) = p(b) = (u,v)$ but $a≠b$.
If $u=v$, I can show that there are at least $3$ points using the same technique. I tried to extend this method by showing that the adherence of the net $([x_i:y_j])_{i,j} ⊆ ℝP^1$ contains many points but this is actually false.
What is the cardinality of $p^{-1}(u,v)$ (assuming AC and CH)?
The cardinality of $p^{-1}(u,v)$ depends on the particular ultrafilters $u$ and $v$. It can be any finite number (since you've assumed CH) or $2^{2^{\aleph_0}}$. Smaller infinite cardinals are impossible because every infinite closed subset of $\beta\mathbb N$ (or $\beta(\mathbb N^2)$) includes a copy of $\beta\mathbb N$ and therefore has cardinality $2^{2^{\aleph_0}}$.
Details about the connection between the cardinality of $p^{-1}(u,v)$ and the combinatorial properties of $u$ and $v$ are in a paper I wrote with Gugu Moche, available at http://www.math.lsa.umich.edu/~ablass/moche.pdf .