I've heard everything from zero-dimensional points, to squares, and I would love to know what they really look like, or if they have any physical shape.
In physics quarks are point elementary particles. Their existence can only be inferred by fitting mathematical equations derived from quantum mechanical solutions of differential equations.
A simplified description can be found here.
Due to a phenomenon known as color confinement, quarks are never directly observed or found in isolation; they can be found only within hadrons, such as baryons (of which protons and neutrons are examples), and mesons. For this reason, much of what is known about quarks has been drawn from observations of the hadrons themselves.
The most dominant extension of present day theories gives the "shape" of a closed string to elementary particles, i.e. fits them as excitations of one dimensional objects following a string theory, but the "point" is a good approximation since these strings have very small dimensions.
Levels of magnification: 1. Macroscopic level: Matter 2. Molecular level 3. Atomic level: Protons, neutrons, and electrons 4. Subatomic level: Electron 5. Subatomic level: Quarks 6. String level
Please note that for clarity the standard model particles are shown as little balls, event though they are points at that level of magnification.
I've not studied this at all but I have a theory that they are shaped like a triangular pyramid, all four sides being equilateral. It is the least amount of lines to create a three dimensional object, and being the simplest form of matter, it would be simple... But like I've said, this is just an unstudied guess