A single hydrogen atom has only one electron. Thus, it is in an (electronic) doublet state and there are no singlet or triplet states. However, this is only the case if one neglects the spin of the nucleus. The spin of the nucleus can be parallel or anti-parallel to the spin of the electron leading "triplet-like" or "singlet-like" states. The Bohr model does not account for this effect and it is what your book refers to. This is known as the Hyperfine structure.
However, if one talks about the triplet and singlet states of hydrogen one usually means molecular hydrogen $H_2$. This molecule has two electrons and is thus either a singlet or a triplet, but nuclear spin is important here, too: https://en.wikipedia.org/wiki/Spin_isomers_of_hydrogen