Why is there no neutron in protium?
Your chemistry teacher is making a few simplifications there that make the statement false on a black-and-white true-and-false scale. Protons would repel each other electrostaticly due to their same charges. Neutrons interact with protons by the so-termed strong interaction (because it is stronger than the weak interaction; props to physicists for inventing names) which lead to stability of the nucleus, preventing it from falling apart due to the repulsion.

Hydrogen, specifically the $\ce{^1H}$ isotope, does not need neutrons to stabilise the nucleus because one proton cannot repel itself.

Note that ‘electrons crashing into the nucleus’ is a very strong simplification that you should eliminate from your language. Electrons are waves for all that matters so they are all around the nucleus at the same time. In fact, the highest probability of detecting them is right there, in the nucleus. (On a simpler scale, Bohr once invented ‘orbits’ that you can rationalise much like the orbit of a satellite, which also does not crash onto Earth. But don’t tell a quantum chemist or nuclear physicist that I said that!)

Finally, it is wrong to say ‘at least as many neutrons as protons’. $\ce{^3He}$ is a stable isotope with less neutrons than protons in the nucleus.