We　propose　an　effective　protocol　for　the　implementation　of　nonadiabatic　geometric　quantum　gates　of　cat-state　qubits　in　Kerr-nonlinear　resonators　driven　by　two-photon　squeezing　drives.　Coupling　the　Kerr-nonlinear　resonators　with　an　auxiliary　qutrit　with　proper　coupling　strengths,　the　selective　transition　of　the　auxiliary　qutrit　is　realized.　The　selective　transition　can　be　exploited　in　the　implementation　of　a　set　of　useful　quantum　gates,　including　the　phase　gates,　the　NOT　gates,　the　controlled-phase　gates,　the　controlled　NOT　gates,　and　the　Toffoli　gates.　Numerical　simulations　show　the　implementations　of　different　types　of　gates　are　robust　against　systematic　errors,　random　noise,　and　decoherence.　Therefore,　the　protocol　may　be　helpful　for　robust　and　scalable　quantum　computation　based　on　cat-state　qubits.　©　2023　The　Author(s).　Published　by　IOP　Publishing　Ltd　on　behalf　of　the　Institute　of　Physics　and　Deutsche　Physikalische　Gesellschaft.