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As I understand, particles such as the neutron, whilst having no external charge still possess a magnetic moment due to the underlying charges of its components.

By that logic why does the alpha particle have a magnetic moment of zero?

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No the neutron has no charge but it has a magnetic moment due to the rotation of the charges that compose it. – Christopher Yeates Jan 25 at 19:27

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This is actually the least hypothesis for this configuration.

Four nucleons comprising all four allowed $\text{spin} \times \text{isospin}$ states can all be expected to be in the s1 state so that they have no orbital contribution and for each pair the intrinsic magnetic moments cancel as the spins are opposed. Boom. Zero and done.

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a more interesting question is how the spherical symmetry in the ground state of the $He_4$ nucleus is determinant for the degeneracy observed in the Bose-Einstein condensate, and why similarly symmetric nuclei do not show the condensate transition – lurscher Jan 25 at 19:59
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@lurscher Can bosons that are composed of several fermions occupy the same state? has some discussion on that issue, but it is mostly beyond my poor, weak, experimentalist brain. – dmckee♦ Jan 25 at 20:04

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