We report on Swift X-ray Telescope observations of Eta Carinae ({eta}Car), an extremely massive, long-period, highly eccentric binary obtained during the 2014.6 X-ray minimum/periastron passage. These observations show that {eta} Car may have been particularly bright in X-rays going into the X-ray minimum state, while the duration of the 2014 X-ray minimum was intermediate between the extended minima seen in 1998.0 and 2003.5 by Rossi X-Ray Timing Explorer (RXTE), and the shorter minimum in 2009.0. The hardness ratios derived from the Swift observations showed a relatively smooth increase to a peak value occurring 40.5 days after the start of the X-ray minimum, though these observations cannot reliably measure the X-ray hardness during the deepest part of the X-ray minimum when contamination by the "central constant emission" component is significant. By comparing the timings of the RXTE and Swift observations near the X-ray minima, we derive an updated X-ray period of P_X_=2023.7+/-0.7days, in good agreement with periods derived from observations at other wavelengths, and we compare the X-ray changes with variations in the HeII4686 emission. The middle of the "Deep Minimum" interval, as defined by the Swift column density variations, is in good agreement with the time of periastron passage derived from the HeII{lambda}4686 line variations.