Assume $C$ and $\mathbb{R}_{\ge 0}^n$ can be (non-strictly) separated by a subspace of dimension $n-1$. Then a normal vector $x$ to that subspace lies in $\mathbb{R}_{\ge 0}^n$; see e.g. [here][1] for a proof.

But then by your asummption, there exists $y \in C$ on the same side of this subspace as $\mathbb{R}_{\ge 0}^n$, contradiction.


  [1]: https://math.stackexchange.com/questions/106312/intersection-between-orthogonal-complement-of-a-subspace-and-a-set