Unlike other elements, why doesn't hydrogen have even a single neutron?

All the elements, except hydrogen, have at least 1 neutron in their nucleus. In fact, this statement is only partially correct. There are isotopes of hydrogen that have neutron/s. The isotope, protium, does not have any neutron, but deuterium has 1 neutron and tritium has 2 neutrons.

Neutrons are the stabilizers in the nucleus of an atom. The nucleus of almost all the elements consist of protons and neutrons. Protons are positively charged particles and as such will repel each other (since same charges repel each other); for that reason, the nucleus would not be stable if there were only protons in it. The presence of neutrons stabilize the nucleus. Neutrons add strong forces to the nucleus, which are sufficient to overcome the electrostatic repulsion of the protons, thereby stabilizing the nucleus. This is the reason that most of the elements have about the same number of protons and neutrons in them.

In the case of hydrogen, there is only 1 proton (for protium) and hence there is no need of a neutron. This is the reason why, unlike other elements, the hydrogen atom (protium atom) does not have a neutron.