This is false for all $n \geq 2$, but true for $n=1$.
A map $g \colon X \to Y$ of topological space is said to be free if there is a map $f \colon X \to X$ such that $g(x) \neq g(f(x))$ for all $x\in X$. This definition appears to be due to Hopf, and was given in
H. Hopf, Freie Überdeckungen und freie Abbildungen, Fund. Math. 28 (1937), 33--57.
In that same paper, Hopf apparently proved that:
- There is a free map $g \colon S^n \to \mathbb{R^n}$ for every $n \geq 3$.
- There is no free map from $S^n$ into $\mathbb{R}$ if $n \geq 2$.
A generalization of (1) was obtained by Pannwitz in
E. Pannwitz, Eine freie Abbildung der n-dimensionalen Sphäre in die Ebene, Math. Nachr. 7 (1952), 183–-185.
According to the MR, she proved that, for every $n \geq 0$, there exists a free map $g \colon S^n \to \mathbb{R}^2$.
Unfortunately, I'm having trouble accessing the papers, so I can't provide any more details.