Lerch-$\small \zeta(\varphi,0,-n)$ of integer *n* purely real and imaginary ($\small \zeta_\varphi (-n)^2 $ is real) for $\small n \ge 2$?

By the symmetry of the triangle of Eulerian numbers the rational function $$R_{n}(z) = {z \over (1-z)} \cdot E(z,n) \cdot (1-z)^{-n} $$ satisfies$$R_{n}(z)=(-1)^{n+1}R_{n}(\frac{1}{z}).$$ Since the numerator and denominator polynomials of $R_{n}(z)$ have real (in fact, integer) coefficients we have$$\overline{R_{n}(z)}=R_{n}(\bar z).$$ Your observation should now be explained (there is no need to restrict to $n>2$).

To clarify further:

Let $ z = \exp(i \cdot \varphi) \ne 1 $ lie on the unit circle so $\bar z = \frac {1}{ z}$ and let $n \geq 1$. Suppose $R_{n}(z) = a+ib$.

Then from the above$$\overline{R_{n}(z)}=R_{n}(\bar z) = R_{n}(\frac{1}{z})=(-1)^{n+1}R_{n}(z). $$ Hence $$a-ib = (-1)^{n+1}(a+ib).$$ If $n$ is even we get $a = 0,$ and so $R_{n}(z)$ is pure imaginary; if $n$ is odd we get $b= 0,$ and $R_{n}(z)$ is real.

A particularly interesting case is when $z = i$. The sequence $(2 \cdot R_{n}(i))n \geq 1$ equals $[-1, -1 \cdot i, 2, 5 \cdot i, -16, -61 \cdot i,...]$, which is the sequence of up-down numbers multiplied by powers of $i$. For more on this see my comment of Jan 2011 in A000111 in the OEIS.