Abstract An Earth-analog orbiting within the habitable zone of α Centauri B was shown to undergo large variations in its obliquity, or axial tilt, which affects the planetary climate by altering the radiative flux for a given latitude. We examine the potential implications of these obliquity variations for climate through Milankovitch cycles using an energy balance model with ice growth and retreat. Similar to previous studies, the largest amplitude obliquity variations from spin-orbit resonances induce snowball states within the habitable zone, while moderate variations can allow for persistent ice caps or an ice belt. Particular outcomes for the global ice distribution can depend on the planetary orbit, obliquity, spin precession, binary orbit, and which star the Earth-analog orbits. An Earth-analog with an inclined orbit relative to the binary orbital plane can periodically transition through several global ice distribution states and risk runaway glaciation when ice appears at both poles and the equator. When determining the potential habitability for planets in general stellar binaries, more care must be taken due to the orbital and spin dynamics. For Earth-analogs within the habitable zone of α Centauri B can experience a much greater range of climate states, which is in contrast to Earth-analogs in the habitable zone of α Centauri A.