We search for the flavor-changing neutral-current decays B→K^(*)νν, and the invisible decays J/ψ→νν and ψ(2S)→νν via B→K^(*)J/ψ and B→K^(*)ψ(2S), respectively, using a data sample of 471×10^6 BB pairs collected by the BABAR experiment. We fully reconstruct the hadronic decay of one of the B mesons in the Υ(4S)→BB decay, and search for the B→K^(*)νν decay in the rest of the event. We observe no significant excess of signal decays over background and report branching fraction upper limits of B(B^+→K^+νν)