We report measurements of the branching fractions and CP asymmetries for B^+- -> eta h^+- (h = K or pi) and the observation of the decay B^0 -> eta K^0 from the final data sample of 772x10^6 BBbar pairs collected with the Belle detector at the KEKB asymmetric-energy e^+e^- collider. The measured branching fractions are Br(B^+- -> eta K^+-) = (2.12 +- 0.23 +- 0.11)x10^-6}, Br(B^+- -> eta pi^+-) = (4.07 +- 0.26 +- 0.21)x10^{-6} and Br(B^0 -> eta K^0) = (1.27^{+0.33}_{-0.29} +- 0.08)x10^-6, where the last decay is observed for the first time with a significance of 5.4 standard deviations (σ). We also find evidence for CP violation in the charged B modes, A_{CP}(B^+- -> eta K^+-) = -0.38 +- 0.11 +- 0.01 and A_{CP}(B^+- -> eta pi^+- = -0.19 +- 0.06 +- 0.01 with significances of 3.8 sigma and 3.0 sigma, respectively. For all measurements, the first and second uncertainties are statistical and systematic, respectively. ; Comment: 6 pages, 3 figures, 1 table