We report the results of a study of the exclusive charmless semileptonic decays, B+→η(′)ℓ+ν and B0→π-ℓ+ν, undertaken with approximately 464×106 BB̅ pairs collected at the Υ(4S) resonance with the BABAR detector. The analysis uses events in which the signal B decays are reconstructed with a loose neutrino reconstruction technique. We obtain partial branching fractions for B+→ηℓ+ν and B0→π-ℓ+ν decays in three and 12 bins of q2, respectively, from which we extract the f+(q2) form-factor shapes and the total branching fractions B(B+→ηℓ+ν)=(0.36±0.05stat±0.04syst)×10-4 and B(B0→π-ℓ+ν)=(1.42±0.05stat±0.07syst)×10-4. We also measure B(B+→η′ℓ+ν)=(0.24±0.08stat±0.03syst)×10-4. We obtain values for the magnitude of the CKM matrix element |Vub| using three different QCD calculations.