"Later in the book it talks about motion in a centrally symmetrical field and says that Dirac's equation cannot be used for r tending to zero."
That is not quite what the authors mean. If you look ahead in the same chapter, you'll find §36 Motion in a Coulomb field, and the well-behaved solutions there tell you that Dirac's eq. can be used just fine with a Coulomb field. It actually produces the Sommerfeld fine-structure formula for the bound energy levels in Coulomb field.
Your other paragraph discusses the behavior of the solutions to the radial part of the Dirac eq. as $r \rightarrow 0$ "assuming that the field $U(r)$ increases more rapidly than $1/r$ as $r \rightarrow 0$". The argument is that for any potential stronger than a Coulomb field there will always be negative energy bound states, which imply "that the system is unstable with respect to the generation of electro-positron pairs".
"Where in the formalism does it predict/mandate the creation of this electron and positron pair? Which equation says that?"
The screening of the Coulomb potential by electron-positron generation or vacuum polarization was first calculated by Uehling in 1935, see Uehling potential.
If we keep strictly within the bounds of relativistic QM and do not go into QED, the bit about "the generation of electron-positron pairs" follows from the electron-hole interpretation of the Dirac states, plus the CPT symmetry guaranteed by the theorem by the same name. In hole theory electron states, including bound ones, always have positive energy, and "charge conjugation symmetry" means that negative energy states are indistinguishable from positive energy states of the charge-conjugated particle - the positron. The vacuum state is a fully occupied, symmetrized sea of electrons and positrons.
By this interpretation, an electron in a bound state of negative energy under an attractive potential is indistinguishable from a positron in a positive energy state in the same potential. The "scattering" of the negative energy electron into a positive energy positron and conversely perturbs the VEV of the charge density around the nucleus (vacuum polarization) which in turn screens the Coulomb potential (Uehling potential). It is sometimes said that bound states of negative energy are indicative of a "vacuum collapse".