A neutral atom is an atom with an equal number of protons and electrons
We know the force of attraction between the proton and electron is equal and assuming that:
How can a neutral atom attract electrons when it's supposed to have zero charge ?
An atom is only neutral when viewed as a single object from large enough distance. But as an electron comes closer to the atom, it "notices" the electron cloud first. This cloud also "notices" the electron and deforms—the atom polarizes—so as to keep the atomic electrons farther on average from the extra electron, since like charges repel. But this deformation leads to less shielding of the nucleus than in isolated atom, so the extra electron begins to feel a non-negligible attracting electric field.
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.
The electronegativity of atoms/elements is indeed an indication of the relative attractiveness of that atom/element for an electron.
But it seems to me that the question asks what force or energy is involved. In general, atoms and molecules are neutral. It takes energy to produce a free electron and hang it in space. If it detects a positive charge, it will be attracted and become stabilized by emitting energy as photons or heat. A less stabilizing destination for an electron would be a "non-bonding" or slightly anti-bonding orbital - this is electronegativity. it's just a better (more stable) place for an electron than plain old free space. And the force of attraction can be measured by the energy required to remove the electron from thin new place (whether it is a really good place or just an OK place - but still better than free space).
The answer lies in electronegativity. When a proton attracts an electron, the electron doesn't magically suck out the charge of the proton. The proton's charge is still distributed in all directions. The reason why 1 proton on average can attract only 1 electron is because electrons push each other out.
Now let's first take H - it has 1 proton which attracts 1 electron. If another electron jumps in, only 1 electron stays in the end. Then He - it has 2 protons, so it attracts electrons even more. So even though electrons are fighting for the place, the nucleus charge is enough to hold them.
It gets interesting with Li. It should have 3 electrons, right? But 1st shell can take only 2 electrons, so the 3d electron must go to the 2nd shell which is further away. In such case the inner shell of electrons has a much greater effect on the outer electron, this is called electron screening, not to mention that the further you are from nucleus - the weaker the attraction. So even though Li has more protons than He, it's too weak to hold electrons on the 2nd shell, so some other atom will take the electron away and Li will be ionized and become Li$^+$ with only 2 electrons.
How strongly an atom wants new electrons is called electronegativity. It increases to the right of the periodic table because nucleus gets larger and larger and can hold on more and more electrons. In the last columns atoms want electrons so much that they can mug other atoms with weaker electronegativity.
But then the row of the table finishes and new row starts. At this point previous shell is completely filled and a new shell starts, and the electron screening kicks in again.
You can see these trends here. Electronegativity is the reason why Na & Cl can't form a molecule (covalent bond) - Cl (strong electronegativity) simply takes Na's (weak electronegativity) electron and both become ions: Cl$^-$ and Na$^+$. In the end they form an ionic bond instead.
