If I'm reading your assumptions correctly, any compact set $E$ whose Hausdorff dimension is less than $n - 1$ but whose packing dimension is greater than $n - 1$ would be a counterexample. But I could be misinterpreting a quantifier.
Here's the intuition. Obviously we puny humans can't see the infinitesimal world, but things still appear $1$-dimensional, $2$-dimensional, or $3$-dimensional to us. Imagine you're nearsighted and can't tell the difference between two points at scale $r$. Let's informally say that the "dimension of $E$ at scale $r$", $s_E(r)$, is what dimension you perceive $E$ to have. Then the Hausdorff dimension is basically $\liminf_{r \to 0} s_E$ while the packing dimension is basically $\limsup_{r \to 0} s_E$. Obviously none of this is rigorous, but sometimes you hear measure theory people talking about "algorithmic fractal dimension" or "branching functions" which are both ways of making this intuition precise. Anyways, the point is we need a set which at some scales looks like it has small dimension and at other scales looks like it has big dimension.
Let $n = 2$ and start with $E_0 := [0, 1]^2$. Now let $E_1 = [1/3, 2/3] \times ([0, 1/3] \cup [2/3, 1])$, $E_2 = [4/9, 5/9] \times ([0, 1/9] \cup [2/9, 3/9] \cup [6/9, 7/9] \cup [8/9, 1])$, et cetra, so we're building a Cantor set in the line $\{1/2\} \times \mathbb R$. Keep doing this until you've built $E_{100}$, which is a union of cubes of side length $3^{-100}$. Break up each such cube into $9$ subcubes of side length $3^{-101}$ and remove the middle one, as in the construction of the Sierpinski carpet. Then do this for another $100$ stages. You get cubes of side length $3^{-200}$, in each one build a Cantor set in a line for $100$ stages, until you get cubes of side length $3^{-300}$, then in each of those build a Sierpinski carpet, and so on...
The Hausdorff dimension of this set is $\log 2/\log 3 < 1$, because when you try to compute the $s$-dimensional Hausdorff measure where $s > \log 2/\log 3$, you can bound that from above at each scale where $E$ looks like a Cantor set, and at those scales the measure is going to look arbitrarily small. But, in the definition of packing dimension, you need to be able to pack this set with balls at EVERY scale, and that includes the scales where it looks like a Cantor set. So the packing dimension of this set is $\log 8/\log 3 > 1$.