© 2015 American Physical Society. We study the structure of the eigenstates and the dynamics of a system that undergoes an excited-state quantum phase transition (ESQPT). The analysis is performed for two-level pairing models characterized by a U(n+1) algebraic structure. They exhibit a second-order phase transition between two limiting dynamical symmetries represented by the U(n) and SO(n+1) subalgebras. They are, or can be mapped onto, models of interacting bosons. We show that the eigenstates with energies very close to the ESQPT critical point, EESQPT, are highly localized in the U(n) basis. Consequently, the dynamics of a system initially prepared in a U(n)-basis vector with energy E?EESQPT may be extremely slow. Signatures of an ESQPT can therefore be found in the structures of the eigenstates and in the speed of the system evolution after a sudden quench. Our findings can be tested experimentally with trapped ions.