An indirect argument:

Since the Banach space of continuous functions $C(\beta\mathbb{N})$ is isomorphic to $\ell_\infty$, it contains no complemented copies of $c_0$.

Since $C(\beta\mathbb{N}\times\beta\mathbb{N})$ is isomorphic to $C\big(\beta \mathbb{N},C(\beta\mathbb{N})\big)$, it contains a complemented copy of $c_0$. See [P. Cembranos. $C(K,E)$ contains a complemented copy of $c_0$. Proc. Amer. Math. Soc. 91 (1984), 556-558.]

An indirect argument:

Since the Banach space of continuous functions $C(\beta\mathbb{N})$ is isomorphic to $\ell_\infty$, it contains no complemented copies of $c_0$.

Since $C(\beta\mathbb{N}\times\beta\mathbb{N})$ is isomorphic to $C\big(\beta \mathbb{N},C(\beta\mathbb{N})\big)$, it contains a complemented copy of $c_0$. See [P. Cembranos. $C(K,E)$ contains a complemented copy of $c_0$. Proc. Amer. Math. Soc. 91 (1984), 556-558.]