Write $[a,b] = [a^{(1)},b^{(1)}] \times \ldots \times [a^{(n)},b^{(n)}]$ and $A \Delta B = A \setminus B \cup B \setminus A$ for the symmetric difference.
Consider $x_0=(x_0^1,\dots,x_0^n)$ and $u_0=(u_0^1,\dots,u_0^n), u=(u^1,\dots,u^n) \in B(x_0,\delta)$ and $\lambda$ the Lebesgue's measure in $\mathbb{R}^{n}$. Is it true that
$$\lambda([x_0,u]\Delta [x_0,u_0]) \leq C \sum_{i=1}^{n}|u^i-u_0^i|$$ for some $C>0$?
I started by considering $x \in [x_0,u]\Delta [x_0,u_0]$. W.L.O.G we can suppose that $x \in [x_0,u]\setminus[x_0,u_0] $. Therefore, $x_0^i \leq x^i \leq u^i$ and there exists $i_0 \in \{1,\dots, n\}$ such that $x^{i_0} \not\in [x_0^{i_0},u_0^{i_0}]$. Thus, $x^{i_0}>u_{0}^{i_0}$, which implies that $u_0^{i_0}<x^{i_0}\leq u^i$. The problem is that this doesn't provide any information for the other coordinates that is, for $i \neq i_0$.