A neutral atom is an atom with an equal number of protons and electrons [...] The "neutral" in a neutral atom means electrically neutral.
Neutral in this case means net neutral. It does not mean that the opposite charges "destroy" each other. After all, the electrons are still attracted to the nucleus even though the overall system is net neutral.
We know the force of attraction between the proton and electron is equal [...]
I am unsure what that statement is supposed to mean. Maybe it means a proton and an electron attract each other, i.e. both experience a force? That is certainly true.
one proton's positive charge attracts one electron
That is not quite true. The charge of the proton attracts any number of particles with negative charge simultaneously.
How can a neutral atom attract electrons when it's supposed to have zero charge?
It has a net zero charge. Irrespective of the net charge of an atom or ion, when an electron approaches, it is attracted by the nucleus and repulsed by the electrons that are already there. Depending on which force is larger, it will stay or go.
This so far is a non-quantum argument, but I think it is sufficient to address some of the misconceptions. You have to consider quantum effects to explain why the electrons don't fall into the nucleus, and why all the electrons are in distinct states.
What else can we learn from this?
Neutral particles such as helium atoms do attract each other (dispersion forces). Anions (such as fluoride) do form and are stable in the gas phase. Neutral molecules can be polar, and bind to anions.