We study theoretically the bound state spectrum and 0-pi transitions in ballistic quasi-one-dimensional superconductor/ferromagnetic insulator/superconductor Josephson junctions. In addition to the Andreev bound states, stemming from the phase coherence, the magnetic barrier gives rise to qualitatively different Yu-ShibaRusinov (YSR) bound states with genuine spectral features and spin characteristics. We show that zero-energy YSR states are much more robust against the presence of scalar tunneling than their Andreev counterparts and also fingerprint a quantum phase transition from the junctions' 0 into the pi phase, connected to a measurable reversal of the Josephson current; this evidence persists also in the presence of Rashba spin-orbit coupling.