So hydrogen can be stable with no neutrons. Helium requires 2 neutrons in order to be most stable, oxygen requires 8, and so on.
So what is it about neutrons that stabilize atoms?
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There are two primary forces relevant to nuclear stability. Electromagnetic charge, and the nuclear strong force. Electromagnetic force attracts opposite charges and repels similar charges (like pushing the “wrong” sides of two magnets together). The strong force pulls all particles together, but only works on really small distances.
In a large atom the nucleus has lots of protons. They really don’t want to stay together because they’re all positively charged. Neutrons spread out the protons, effectively diluting the EM repulsion, allowing the strong force to keep it together.
A hydrogen atom needs no neutrons at all because it only has a single proton.
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