I need to compute the Hausdorff dimension of certain sets using a computer and, to date, my approach has been to use a Box-counting algorithm, for I once read that the Hausdorff dimension of an auto-similar fractal set matches its Box-counting dimension. Now, my sets are not completely self-similar because they include a random factor on their construction, though the seed is always an auto-similar fractal set. Hence I was wondering if there is some wider class of sets for which both dimensions coincide. I tried proving such property for compact sets but I soon found a counter-example (namely the set $X=\left\{1/n, n \in \mathbb{N} \right\} \cup \left\{ 0 \right\}$). I thought that maybe connectedness may play a role, but to the point I haven't found any piece of literature regarding this or any similar facts.
So, any knowledge on the topic or known references would be most appreciated.
Do not expect topological properties (compactness, connectedness, etc) to tell you much about metric properties (such as the Hausdorff dimension). To get an idea of how difficult the question of equality is, consider the following: the graph of Weierstrass function $y=\sum a^k \cos (2\pi b^k x)$ with $ab>1$, $a<1<b$, has been known to have Minkowski dimension $D=2+\frac{\log a}{\log b}$ since 1984. Its Hausdorff dimension is conjectured to also be equal to $D$, but this remains an open problem despite a large number of papers published on the subject.
One class of sets in which Minkowski and Hausdorff dimensions coincide is Ahlfors regular sets (also known as Ahlfors-David regular, or simply regular). These sets support a measure $\mu$ such that $\mu(B)\approx (\operatorname{diam} B)^d$ for every ball $B$. The catch is: if you can prove that a set is Ahlfors regular, then you already know its Hausdorff dimension, namely $d$.
Since you already know Minkowski dimension $d$, I think that your best bet is to look for a positive measure $\mu$ supported on the set that has finite energy $\iint |x-y|^{-t}\,d\mu_x\,d\mu_y$ for $t<d$. The existence of such a measure (for every $t<d$) implies that the Hausdorff dimension is at least $d$. In 1998 Hunt used this approach to show that the graph of a random Weierstrass-type function has the conjectured Hausdorff dimension $D=2+\frac{\log a}{\log b}$ with probability $1$. See The Hausdorff dimension of graphs of Weierstrass functions. Heuristically speaking: for the Hausdorff and Minkowski dimensions to be different, the set must have infinitely many "abnormal" scales; and for a random set the probability of this happening is $0$. Of course, how true the preceding sentence is depends on the kind of randomness you have.
I also recommend the recent paper On the dimension of the graph of the classical Weierstrass function by Baranski, Barany and Romanowska; it includes a survey of recent results, including ([1] and [7] in the paper) the non-equality of Minkowski and Hausdorff dimensions of the graph of a sparse-scale Weierstrass-type function.