The following is not a proof of the statement, sorry. But it contains one correct idea

Let us associate to our function $F$ the curve $C$ in unite $S^{n-1}$, given by the vector $\frac{F}{|F|}$. Then it follows from $F(0)=F(1)$ that the convex hull of the curve $C$ containes the point $0$. It follows that there is a simplex with vertices on $C$ that containes $0$. If we could find such a simplex, that has $n$ vertices, we will be done. But it is not sure. Sorry for previous claiming that this proves the statement, Ilia is correct.