Re: Why do proton and neutrons have spin one-half?

Well, this has both a simple answer and a complicated answer. The simple
answer is that in the proton and neutron the three quarks align themselves
so that two of them have their spins in one direction and one is in the
other direction (two up and one down), so that the net spin is one-half.
This appears to be the lowest-energy configuration (there are particles
made of the same three quarks which do have spin three-halves but they are
heavier and decay very quickly).

The complicated answer is that there are other things in the proton and
neutron besides the three quarks; there are gluons (which have spin 1),
which the quarks exchange as they stay bound together, and also virtual
quark-antiquark pairs. Recent work on the spin structure of the proton has
shown that, despite the nice, simple picture in the first paragraph, most
of the intrinsic angular momentum (or "spin") of the proton does not in
fact come from the quarks! So where does it come from? Well, presumably from
some combination of the gluons and virtual quark-antiquark pairs; at the
moment, though, this is a mystery which is still under investigation.

You can find a good introduction to the basic ideas of particle physics
at The Particle Adventure, http://pdg.lbl.gov/cpep/adventure_home.html.