Examples of compact complex non-Kähler manifolds which satisfy $h^{p,q} = h^{q,p}$

I believe (he said with some trepidation) that the results of Angella and Tomassini (which say that this property is preserved under complex analytic deformations) give lots of examples. This is in Angella's thesis, and in their joint Inventiones article (2012) - they also discuss at length the obstructions (or lack thereof) to the existence of such deformations.

Complete smooth toric varieties $X_\Sigma$, corresponding to regular non-polytopal fans $\Sigma$ form a large series of manifolds bimeromorphic to Kahler, but non-Kahler.

At the same time, their Hodge diamond is concetrated on the main diagonal: $h^{p,q}=0$, unless $p=q$.

The smallest possible dimension of such an example is 3: see Fulton's "Introduction into toric varieties", Excersise in Section 3.4.

Every compact complex manifold satisfying the $\partial\overline{\partial}$-Lemma has such a property. Particular examples are given by - as you already said - Hironaka example (and, more in general, Moishezon manifolds and manifolds in class C of Fujiki), or some deformations of twistor spaces (see LeBrun, Poon, Twistors, Kähler manifolds, and bimeromorphic geometry. II, J. Amer. Math. Soc. 5 (1992), no. 2, 317–325).

On the other side: in Ceballos, Otal, Ugarte, Villacampa, arXiv:1111.5873, Proposition 4.3, you find a concrete example of a compact complex manifold with the symmetry of Hodge diamond you require. This example does not satisfy $\partial\overline{\partial}$-Lemma.