Brain science fact from my thesis research: the EEG signal doesn't actually come from your neurons firing! Because of the way action potentials are structured, their effects on the electric field cancel themselves out by the time they make it outside of your head. The EEG actually comes from your dendrites, which are the tree-like parts of the neuron that collect signals from other neurons. But even then, only some neurons have the right shape for their dendrites to contribute to the EEG.
So first a quick explanation of how a neuron works: you can divide a neuron into three parts: dendrites, soma, and axon. The dendrites are a bunch of branching projections that recieve signals from other neurons. They turn those signals into electrical potentials, which pass down the dendrites and collect in the soma, which is the body of the cell. When enough potential collects in the soma, it sends a signal down the axon, which is a long arm that stretches to the dendrites of other neurons. That signal is called an action potential, and it's what we talk about when we say that a neuron fires.
The thing about action potentials is that they don't work like sending current through a wire. Axons can be long, and they want to convey signals quickly. But the electric potential in a neuron comes from different concentrations of ions. If you just push ions into one end of the axon and wait for them to diffuse to the other end, it could take too long and you'd have to worry about the ions leaking through the membrane. (Leaking is why white matter exists - it's basically a bunch of axons that are covered in fat so they don't leak as much.)
So instead of letting the ions just diffuse through the axon, you have action potentials. When a small part along the axon sees the electric potential go up enough, it lets in a bunch of positive ions. That triggers the next part along the axon to do the same thing. Then it lets in a bunch of negative ions for a while to stop the action potential from moving backwards through the axon as well as forwards. So think people in a baseball stadium doing the wave, instead of someone walking from one end of the stands to the other.
But that means that whenever you have positive ions flowing into the axon, there are negative ions flowing in right behind it. You don't get a big difference in charge between one end of the axon and the other. So when you look at the electric field from far away, through your scalp, the positive and negative charges blur together and you don't see anything.
Dendrites, on the other hand, do work kind of like current flowing through a wire. You put ions in on one end, and they diffuse to the other end. So that does create a signal that you can see in the EEG. But if you just have a bunch of dendrites pointing in random directions, they'll still cancel each other out. You get the best signal when you have a lot of dendrites all pointing in the same direction. And the main source of that is certain kinds of neurons called pyramidal neurons: specifically a set of them in certain layers of the brain (II, III, and V) that all extend dendrites straight up to the outermost layer.
So an EEG signal is really a representation of the currents in those dendrites, and not of any neurons firing.