How to measure the spin of a neutral particle?

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.

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...