In contrast with the previous incorrect answers that I hadn't noticed, there isn't any ambiguity or confusion about the Bose-Einstein or Fermi-Dirac statistics for composite systems such as atoms.
A particle – elementary or composite particles – that contains an even number of elementary (or other) fermions is a boson; if it contains an odd number, it is a fermion. Bosons always have an integer spin; fermions always have a half-integer spin. By the spin-statistics theorem, the wave function of two bosons is invariant under their exchange but it is antisymmetric under the exchange of two identical fermions. These two rules are theorems for elementary particles and assuming this theorem, it's also trivial to prove these statements for composite particles.
In particular, for a neutral atom, the numbers of protons and electrons are equal so their total parity is even. That's why only neutrons matter. An isotope with an even number of neutrons is a boson (the whole atom: e.g. helium-4); an isotope with an odd number of neutrons is a fermion (the whole atom: e.g. helium-3).
That doesn't mean that composite bosons exhibit all the same physical phenomena like superfluidity that we may observe with some bosons.
The magnetic moment of a charged "elementary enough" particle scales like $1/m$ where $m$ is the mass of the particle. That's why the magnetic moment of the protons, neutrons, and nuclei are about 1,000-2,000 times smaller than the magnetic moments of the electrons. That's why the nuclear spins are largely negligible for the behavior of the atom in a magnetic field.
This is no contradiction because the whole atoms have a much larger magnetic moments than the nuclei separately – because of the neutrons: atoms are not "elementary enough" in this definition. Both the electrons' spins and their orbital angular momentum contribute to an atom's magnetic moment. Also, there exist a higher number of states because an atom is a typical example of the "addition of several angular momenta". The tensor product Hilbert space may be decomposed as a direct sum of Hilbert spaces with fixed values of the total angular momentum. The degeneracy and the magnetic moment of these components depend on the total angular momentum i.e. on the relative orientation of the nucleus-based and electron-related angular momenta.
In effect, the spectral lines of the whole atom exhibit the so-called hyperfine structure. Up to some approximation, the nuclear spin may be totally ignored. But when the considerations from the previous paragraph are properly account for, each spectral line is actually split to several nearby (3 or so orders of magnitude closer to each other) finer spectral lines, each of which corresponds to a different value of the total angular momentum of the whole atom (or, equivalently, a different value of $\vec J_{\rm electrons}\cdot \vec J_{\rm nucleus}$).
It is energetically favourable for diamagnetic moments to anti-align with an applied $B$ field.
In a liquid all these microscopic moments are free to rotate (subject to thermal fluctuations) and as such all anti-align with the applied field.
An example of magnetic moments rotating in a fluid is in the "super paramagnetic" ferrofluid (except here the ferromagnetic nanoparticles align with the $B$ field).
For further reading I recommend "Magnetism and Magnetic Materials" by Coey.
Best Answer
Whenever you have an atomic dipole, in the presence of an external magnetic field, magnetization will occur.
Hydrogen is not usually called paramagnetic since monoatomic hydrogen is very unstable and hydrogen atoms will combine to form H$_2$ (meaning the magnetic moments become quenched - all the spins become paired). Hydrogen (molecular hydrogen) is therefore diamagnetic.
But paramagnetic substances require at least one unpaired electron (since the net magnetic moment of the electrons do not add up to zero). This is because in the presence of a magnetic field, the atomic dipoles align along the field (albeit this magnetization is weak). So atomic hydrogen is in fact paramagnetic because it has an unpaired electron.