I managed to piece together a solution to my problem by reading the first page of this paper http://www-users.math.umn.edu/~bobko001/papers/2010_JMS-165_Conc.on.the.cube.pdf.

I'll only handle the euclidean case $p=2$, as the other cases will follow by equivalence of $\ell_p$-norms (it's possible this "delegation procedure" is not very optimal for...).

So, consider the function $T:\mathbb R^n \rightarrow [0, 1]^n$ defined by $T(z_1,\ldots,z_n)=(\Phi(z_1),\ldots,\Phi(z_n))$ where $\Phi$ is the standard Gaussian CDF. It's easy to see that $u_n=\gamma_n \circ T^{-1}$. We will take for granted that $\Phi$ (and therefore $T$) is $(2\pi)^{-1/2}$-Lipschitz continuous.

Let $A$ be a measurable subset of $[0,1]^n$ and let $B:=T^{-1}A$. Lipschitzness of $T$ implies $A^\varepsilon \supseteq B^{\varepsilon\sqrt{2\pi}}$, and so $u_n(A^\varepsilon) \ge \gamma_n(B^{\varepsilon\sqrt{2\pi}})$. Now, by Gaussian Isoperimetry,
$$
u_n(A^\varepsilon) \ge \gamma_n(B^{\varepsilon\sqrt{2\pi}}) \ge \Phi^{-1}(\gamma_n(B)+\varepsilon\sqrt{2\pi}) = \Phi^{-1}(u_n(A)+\varepsilon\sqrt{2\pi})
$$