How to measure the spin of a neutral particle? If a charged particle with charge $q$ and mass $m$ has spin $s \neq 0$ we can measure an intrinsic magnetic moment $\mu = g \frac{q}{2m}\hbar \sqrt{s(s+1)}$. This is how spin was discovered in the first place in the Stern-Gerlach Experiment.
But for a neutral particle $\mu = 0$, so we cannot measure the spin of the particle in the same manner. But it is said, that e.g. the Neutron or the Neutrino both have a spin $s=1/2$. How was or can this be measured? 
 A: Conservation of angular momentum is invoked for the neutrinos because beams of neutrinos cannot be collimated   for an experimental measurement. Neutron spin  can be measured in a Stern Gerlach setup.
The interactions and decays were carefully examined in various experiments and the only consistent spin values are the ones assigned.
Edit: I see that the question should be formulated as : why the neutron has a Dirac magnetic moment, although it is neutral, which is the formula that is displayed above, and does the neutrino have a Dirac magnetic moment?
The neutron, and other baryons,  has a magnetic moment because the quarks that compose it have a Dirac magnetic moment. See for example Perkins, Introduction to High Energy Physics, section: baryon magnetic moments for the derivation.
Whether the neutrino has a magnetic moment due to higher order loop diagrams is a research question.
So, though spin in charged point like particles is connected to  magnetic moment with the formula above, analogous to classical charges circulating in a loop having a magnetic moment, , charge is not necessary for spin to appear. There is intrinsic spin which for the neutrino comes from the angular momentum balance in the interactions where it appears. The neutrino is a spinor  in the Dirac formalism.
A: Also the neutron has a magnetic moment. Check this out. The reason is that the neutron is not an elementary particle but built up from quarks which have charge...
A: another way to measure 'spin' is through scattering experiments as the scattering cross sections depend on the spin-spin interactions- the most famous example is two types of hydrogen molecules called ortho and para -hydrogen whose scattering with neutrons yielded different cross sections-see any standard textbook on nuclear physics.
I do agree that nucleons/elementary particle's spin quantum states are not related to specific charges on the particles- neutral ones can have 'particle clouds' inside it and they can contribute to the spin state.
some comments atribute spins to the quarks inside the neutron- but spin was available before the 'quarks' were formulated!
