VLA Zeeman observations of absorption lines of neutral hydrogen (H I) and the hydroxyl radical (OH) have been carried out toward the bipolar outflow H II region S106 at a resolution of similar to 20 '' with an rms noise of similar to 3 mJy beam(-1). The results of the observations are interpreted in terms of a model for the absorption-line gas in which some of the OH and H I gas has been accelerated by the passage of a shock front and some of the gas lies outside the shock front at the undisturbed velocity of the molecular cloud. Two distinct OH absorption features are due to gas associated with S106. One component, which is identified with the unshocked gas, is narrow (Delta upsilon(FWHM) = 1.2-2.9 km s(-1)) and is observed at small negative velocities (upsilon(LSR) = -2.4 to -1.0 km s(-1)). This component shows a large peak in opacity at the eastern edge of the H II region near a previously observed molecular cloud core; the H-2 mass of this component is estimated to be similar to 13 M.. In addition, there is a wide (Delta upsilon(FWHM) = 3.7-8.2 km s(-1)) OH velocity component at a higher negative velocity (upsilon(LSR) = -4.8 to -3.3 km s(-1)) that is identified as shocked gas; the H-2 mass in front of S106 from this component is similar to 4 M.. Our observations suggest an enhancement of the OH abundance in both velocity components by a factor of similar to 5 relative to dark clouds; the ratio N-OH/N-H2 approximate to 4 x 10(-7) and N-OH/A(upsilon) approximate to 5 x 10(14). This increased abundance is likely due to the penetration of UV photons into the molecular cloud resulting in enhanced OH formation. A line-of-sight magnetic field (B-los) has been determined for the two OH velocity components between +150 and +400 mu G over the inner arcminute (0.18 pc) of the H II region. The most significant detection of B-los in OH was +400 +/- 23 mu G about 0.5' (0.09 pc) northeast of the exciting source, IRS 4. Given simple assumptions about the geometry of the magnetic field and the mass distribution, the observed magnetic field appears to be near the critical value for magnetic support. Together with far-IR and submillimeter measurements of the direction of magnetic field in the plane of the sky, our Zeeman map suggests that the held may lie along the long axis of S106 at large scales and be twisted into a toroidal morphology near the central star, IRS 4. One of the H I velocity components appears to originate in shocked gas near the ionization front. This H I gas has a high negative radial velocity (upsilon(LSR) = -9.8 to -5.8 km s(-1)), has a large line width (Delta upsilon(FWHM) = 3.9 to 11.6 km s(-1)), and is believed to be from shocked gas associated with the outflow. B-los approximate to +70 +/- 12 mu G for this velocity component near the center of the H II region. The small value of B-los determined for H I relative to OH may be due to a tangling of a strong held near the ionization front on scales smaller than our synthesized beam size. A weak, narrow component with a velocity similar to -2 km s(-1) is observed in the eastern portion of the source; this is probably unshocked H I in the molecular cloud. B-los in this unshocked component is estimated to be between +200 and +300 mu G. The ratio of N-HI/N-H2 x 2 x 10(-2) and 4 x 10(-2) for the unshocked and shocked components, respectively. The increased strength of the shocked H I component is probably due to an increased relative abundance of H I from the photodissociation of H-2.