Compare the boiling points of isobutane and 1-butyne
The boiling points of non-polar hydrocarbons are determined by the extent of van der Waals forces in between them. More specifically, molecules with a larger surface area have larger van der Waals forces of attraction between the molecules.

From the structures, it is obvious that but-1-yne has a larger surface area than isobutane, and hence has the higher boiling point.
Your idea of H-bonding in but-1-yne is plausible, according to this source1:

Quantum chemical calculations indicate hydrogen bond energies in the range $\pu{1–2.2 kcal mol^–1}$. Interconnected contacts such as $\ce{C#C–H ··· C#C–H ··· C#C–H}$ exhibit a pronounced cooperativity effect, with energy increases of $\approx \pu{0.5 kcal mol^–1}$ compared with isolated contacts.

However, while this bonding may be sufficient to explain a higher boiling point compared to isobutane, it is still a very weak bond compared to the H-bonds in hydrogen fluoride or water.
References:

Thomas Steiner, Evgeni B. Starikov, Ana M. Amado, José J. C. Teixeira-Dias, “Weak Hydrogen Bonding. Part 2. The Hydrogen Bonding Nature of Short $\ce{C–H ⋯\pi}$ Contacts: Crystallographic, Spectroscopic and Quantum Mechanical Studies of Some Terminal Alkynes.” J. Chem. Soc., Perkin Trans. 2 1995, (7), 1321–1326 (doi:10.1039/P29950001321).