Definition of the hydrogen bond
In 2011, a modern and more general IUPAC definition of what constitutes a hydrogen bond was recommended (Arunan et al., 2011). The definition is stated as such:
The hydrogen bond is an attractive interaction between a hydrogen atom from a molecule or a molecular fragment $\ce {X-H}$ in which $\ce {X}$ is more electronegative than $\ce {H}$, and an atom or a group of atoms in the same or a different molecule, in which there is evidence of bond formation.
It appears that the definition of then hydrogen bond which includes interactions only with $\ce {F, O or N}$ seems too narrow and restrictive, which I personally feel is indeed the case.
Therefore, it is perfectly fine to say that $\ce {H_2S}$ makes hydrogen bonds with other $\ce {H_2S}$ molecules and that $\ce {HCl}$ makes hydrogen bonds with other $\ce {HCl}$ molecules.
Explaining the boiling points
Wikipedia gives the boiling points of $\ce {H_2S}$ and $\ce {HCl}$ as $\ce {-60 ^{\circ} C}$ and $\ce {-85.05 ^{\circ}C}$ respectively. When rationalising boiling point differences, the first consideration is always the strength of the intermolecular forces between the molecules in the liquid.
The Pauling electronegativities of $\ce {S}$ and $\ce {Cl}$ are $\ce {2.58}$ and $\ce {3.16}$ respectively. Evidently, the electron deficiency in the hydrogen atom bonded to $\ce {Cl}$ is more severe compared to that bonded to $\ce {S}$. This suggests that $\ce {HCl}$ molecules are capable of forming stronger hydrogen bonds. However, it is also important to consider the number of hydrogen bonds formed between molecules, as you have already noted in your question. Each $\ce {H_2S}$ can form 4 hydrogen bonds with other $\ce {H_2S}$ molecules, whereas each $\ce {HCl}$ molecule can only form 2 hydrogen bonds with other $\ce {HCl}$ molecules. Therefore, despite the fact that $\ce {HCl}$ molecules can form stronger hydrogen bonds between themselves, the intermolecular attractions between $\ce {H_2S}$ molecules still prevail as they are twice as numerous. Thus, $\ce {H_2S}$ has a higher boiling point than $\ce {HCl}$.
References
Arunan, E., Desiraju, G. R., Klein, R. A., Sadlej, J., Scheiner, S., Alkorta, I., . . . Nesbitt, D. J. (2011). IUPAC definition of the hydrogen bond. Terminology and nomenclature. Pure Applied Chemistry, 83(8), 1619-1636. doi:10.1351/PAC-REP-10-01-01
Electronegativity. (n.d.). In Wikipedia. Retrieved January 9, 2018, from https://en.wikipedia.org/wiki/Electronegativity
Hydrogen chloride. (n.d.). In Wikipedia. Retrieved January 9, 2018, from https://en.wikipedia.org/wiki/Hydrogen_chloride
Hydrogen sulfide. (n.d.). In Wikipedia. Retrieved January 9, 2018, from https://en.wikipedia.org/wiki/Hydrogen_sulfide
