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 $c_0$](https://doi.org/10.1090/S0002-9939-1984-0746089-2). Proc. Amer. Math. Soc. 
91 (1984), 556-558.]