I think the article is saying that the field affect appears to be completely uniform.
Technically, if the electron can affect something at a great distance via gravity or releasing photons you could say the size of an electron is many light years in diameter. Or you could call it infinitesimally tiny.
Neutrons and protons we know to be complicated little parties of quarks and gluons, each of which are, as far as we know, are also elementary particles like leptons.
Plus I want to point out that it's pointless to say that something is spherical to within 1e-20m unless you also say what the "radius" is. If the radius is 1e-21m it's not a very good sphere at all...
Electrons are point particles - they are true 0 dimensional objects - they have no size at all. However they do have an equivalent wavelength due to their mass, and they have an area of influence.
When you study quantum field theory one of the first mind blowing things that you learn is that an electron at rest can be thought of as a superposition of an infinite number of somewhat classical scenarios. For instance, the electron can shoot of a virtual photon, which creates an electron/positron pair, which annhilate and create a new photon, which is then absorbed by the original electron. These processes can be arbitrarily complex; just imagine substituting in the whole process we just described for the electron that was pair produced- you can do that as many times as you want. These things are all ocurring at once. This leads to the necessity of renormalization and in turn a great joke. What is positive infinity plus negative infinity? If you ask a mathematician he'll tell you that it's undefined. If you ask a physicist he'll tell you it's the mass of the electron.
In between atoms is nothing. Inside atoms (between the electrons and the nucleus) is more nothing. Electrons are point particles and have no size. In between the protons and neutrons of the nucleus is nothing. Inside a proton are quarks, and between those quarks is nothing. Quarks themself are point particles and have no size.
So in total 100% of matter is empty space.
In quantum theory, a particle is described as a probability field that fills space. For example, an electron might have a 2.5% chance of being in some little cube of space, a 10% chance in a nearby cube, and so forth. The rules for how it works are called quantum mechanics.
So the particles are fuzzy. They have no defined size, they can overlap with each other, and so forth.
As usual, this research has been simplified to the point of silliness for the popular press. What they probably mean is that a particular electron orbital in a particular type of atom was measured to be spherically symmetric. That means that they went looking for lumpiness of that electron's probability cloud and found to be perfectly smooth and round.
Particle accelerators have already measured this smoothness at high energies. They crash electrons together at high speed, watch how they scatter off each other, and the scattering statistics are consistent with electrons having no internal bits and pieces. They're just smooth, continuous electron all the way through. (Proton collisions scatter as if there are lots of lumpy bits inside. The bits turn out the be quarks and gluons.)
"Can electrons get squished into different shapes depending on arrangement?"
Yes. While electrons can overlap because their borders are fuzzy, they repel each other in the process, changing each other's shapes.
Even just sitting around in an atom, they often start out various funny shapes because they have wavelike properties: http://en.wikipedia.org/wiki/Atomic_orbital#Orbitals_table