We report a measurement of the flux-averaged neutral-current elastic differential cross section for neutrinos scattering on mineral oil (CH2) as a function of four-momentum transferred squared, Q2. It is obtained by measuring the kinematics of recoiling nucleons with kinetic energy greater than 50 MeV which are readily detected in MiniBooNE. This differential cross-section distribution is fit with fixed nucleon form factors apart from an axial mass MA that provides a best fit for MA=1.39±0.11 GeV. Using the data from the charged-current neutrino interaction sample, a ratio of neutral-current to charged-current quasielastic cross sections as a function of Q2 has been measured. Additionally, single protons with kinetic energies above 350 MeV can be distinguished from neutrons and multiple nucleon events. Using this marker, the strange quark contribution to the neutral-current axial vector form factor at Q2=0, Δs, is found to be Δs=0.08±0.26.