The following claim and its proof answer OP's question, but **only under an additional hypothesis on the critical points** of the coordinate functions $\gamma_i$ of the curve $\gamma = (\gamma_1, \dots, \gamma_n)$.

> **Claim.** Let $n \ge 1$ and let $\gamma = (\gamma_1, \dots, \gamma_n): [0, 1] \rightarrow [0, 1]^n$ be a curve of [class $C^1$](https://en.wikipedia.org/wiki/Smoothness). Assume moreover that $\gamma_i$ has at most countably many [critical points](https://en.wikipedia.org/wiki/Critical_point_(mathematics)) for every $i \in \{1, \dots, n\}$. Then there is $j \in \{1, \dots, n\}$ and $c \in [0, 1]$ such that
$$\vert \{ t \in [0, 1] \, \vert \, \gamma_j(t) = c \} \vert \ge  L(\gamma) /n$$ where $L(\gamma)$ is the [arc length](https://encyclopediaofmath.org/wiki/Rectifiable_curve) of $\gamma$.

I'll be grateful to contributors willing to answer the following question:

**Question.** Is it possible to remove, or at least weaken, the extra hypothesis on critical points?

The proof of the above claim relies on the next two lemmas.

> **Lemma 1.** Let $\gamma = (\gamma_1, \dots, \gamma_n): [0, 1] \rightarrow [0, 1]^n$ be a [rectifiable curve](https://encyclopediaofmath.org/wiki/Rectifiable_curve). Then $\gamma_i$ is rectifiable for every $i$ and there is $j \in \{1, \dots, n\}$ such that $L(\gamma_j) \ge L(\gamma) / n$.

> *Proof.* It follows easily from the definition of the [arc length](https://encyclopediaofmath.org/wiki/Rectifiable_curve) of a curve, and from the inequalities $\vert x_i \vert \le \sqrt{x_1^2 + \cdots + x_n^2} \le \sum_{j = 1}^n \vert x_j \vert$ that 
$$L(\gamma_i) \le L(\gamma) \le \sum_{j = 1}^n L(\gamma_j)$$ for every $i \in \{1, \dots, n\}$. Thus there is $j \in \{1, \dots, n\}$ such that $L(\gamma_j) \ge L(\gamma)/n$.

> **Lemma 2.** Let $\gamma: [0, 1] \rightarrow [0, 1]$ be a curve of class $C^1$. Assume moreover that the set of critical points of $\gamma$ is at most countable. Then we have $$\deg(\gamma) \ge L(\gamma)$$ 
where 
$$\deg(\gamma) := \sup_{y \in [0, 1]} \vert \{ \gamma^{-1}(\{y\})\} \vert.$$

> *Proof.* Since $\gamma$ is of class $C^1$, it is rectifiable, i.e., its arc length $L = L(\gamma)$ is finite.
Thus we can assume, without loss of generality, that $\deg(\gamma) = d < \infty$. Define $s: [0, 1] \rightarrow [0, L]$ by $s(t) = \int_0^t \vert \gamma'(t) \vert dt$. Clearly, the function $s$ is of class $C^1$. Since the set of critical points of $\gamma$ has empty interior, the function $s$ is strictly increasing and is therefore an homeomorphism. Besides, the function $s$ is differentiable at $s(t)$ if and only if $t$ is not a critical point of $\gamma$, i.e., if and only if $\gamma'(t) \neq 0$. Let $\gamma_1 = \gamma \circ s^{-1}$. Then $\gamma_1$ is a continuous curve which is *locally an isometry*, i.e., for every $t \in [0, 1]$ such that $\gamma'(t) \neq 0$, there is an open interval $I = I_t$ containing $s(t)$ such that the restriction of $\gamma_1$ to $I$ is an isometry onto its image. 
Let $O \subseteq [0, 1]$ be the complement of the critical values of $\gamma$. As $O$ is open, it is the union of at most countably many disjoint intervals with non-empty interior. Let $J$ be one of these intervals and denote by $\mathring{J}$ its interior. We claim that $\gamma_1^{-1}(\mathring{J})$ is the union of at most $d$ disjoint intervals which are all  isometric to $\mathring{J}$. It follows from our claim that $[0, L] = \gamma_1^{-1}([0, 1])$ is the disjoint union of an open set of [Lebesgue  measure](https://en.wikipedia.org/wiki/Lebesgue_measure) at most $d$, that is $\gamma_1^{-1}(O)$, and of a set which is at most countable. Therefore $L \le d$. 

> *Proof of the claim.* Combine Lemmas 1 and 2.