A first solution by Cornel Ioan Valean (described in large steps)
From the paper The calculation of a harmonic series with a weight 5 structure, involving the product of harmonic numbers, $H_n H^{(2)}_{2n}$ by Cornel Ioan Valean, we have that
$$I=\int_0^1 \frac{\log^2(1+x)\operatorname{Li}_2(-x) }{x} \textrm{d}x=\frac{2}{15}\log^5(2)-\frac{2}{3}\log^3(2)\zeta(2)+\frac{7}{4}\log^2(2)\zeta(3)-\frac{1}{8}\zeta(2)\zeta(3)\\-\frac{125}{32}\zeta(5)+4 \log (2)\operatorname{Li}_4\left(\frac{1}{2}\right)+4 \operatorname{Li}_5\left(\frac{1}{2}\right).$$
Integrating by parts, we get $I=-5/16\log(2)\zeta(4)+\displaystyle\frac{1}{2}\underbrace{\int_0^1\frac{(\operatorname{Li}_2(-x))^2}{1+x}\textrm{d}x}_{\displaystyle J }$.
Now, the key step (the magical one) is to observe that the Cauchy product of $\displaystyle \frac{(\operatorname{Li}_2(-x))^2}{1+x}$ can be expressed using a powerful sum in the book, (Almost) Impossible Integrals, Sums, and Series, that is the result in $(4.19)$,
$$ \sum_{k=1}^{n-1} \frac{H_k^{(2)}}{(n-k)^2}=\left(H_n^{(2)}\right)^2-5 H_n^{(4)}+4\sum _{k=1}^n \frac{H_k}{k^3},$$
and then we have
$$J=\int_0^1\frac{(\operatorname{Li}_2(-x))^2}{1+x}\textrm{d}x=\sum_{n=1}^{\infty} \int_0^1(-1)^n x^n \left(\left(H_n^{(2)}\right)^2-5 H_n^{(4)}+4\sum _{k=1}^n \frac{H_k}{k^3}\right)\textrm{d}x$$
$$= \sum_{n=1}^{\infty} (-1)^n \frac{\left(H_n^{(2)}\right)^2}{n+1}-5\sum_{n=1}^{\infty} (-1)^n \frac{H_n^{(4)}}{n+1} +4\sum_{n=1}^{\infty} (-1)^n \frac{1}{n+1}\sum _{k=1}^n \frac{H_k}{k^3}.$$
Upon reindexing all series and changing the summation order in the last series (which gives the main series to calculate), everything reduces to using the values of the series $\displaystyle \sum_{n=1}^{\infty}(-1)^{n-1}\frac{H_n^{(2)}}{n^3}$, $\displaystyle \sum_{n=1}^{\infty}(-1)^{n-1}\frac{H_n^{(4)}}{n}$
and $\displaystyle \sum_{n=1}^{\infty}(-1)^{n-1}\frac{(H_n^{(2)})^2}{n}$, which are all given in the book, (Almost) Impossible Integrals, Sums, and Series, on pages $311$ and $529$.
Note that a simple generalization of the series $\displaystyle \sum_{n=1}^{\infty} (-1)^n \frac{H_n^{(4)}}{n}$ may be found in the paper A simple strategy of calculating two alternating harmonic series generalizations by Cornel Ioan Valean.
Full details will be given soon in a new paper.
A bonus of the previous idea: The younger brother of the previous series $\displaystyle \sum_{n=1}^{\infty} \frac{H_n \overline{H}_n}{n^2}$ can be calculated in a similar style. For example, if we use that
$$\begin{equation*}
\sum_{k=1}^{n-1} \frac{H_k^{(2)}}{n-k}=\sum_{k=1}^{n} \frac{H_{k}}{k^{2}}+H_n H_n^{(2)}-2H_{n}^{(3)},
\end{equation*}$$
which you may find in the same book on page $287$, then multiply both sides by $(-1)^n/(n+1)$, sum from $n=1$ to $\infty$, next reverse the Cauchy product in the left-hand side, turn everything into a simple integral and calculate it, together with the fact that for the right-hand side you follow the same arrangements strategy as in the previous series and then make use of the generating functions presented in Sect. 4.10, page $284$ of the mentioned book, to extract the needed alternating series, you're done. Exactly, that simple!
Full details will be given soon in a new paper.
Update: the evaluation of the series appears in the preprint, Two advanced harmonic series of weight 5 involving skew-harmonic numbers.
