Rotationally modulated variability of brown dwarfs and giant planets provides unique information about their surface brightness inhomogeneities, atmospheric circulation, cloud evolution, vertical atmospheric structure, and rotational angular momentum. We report results from Hubble Space Telescope/Wide Field Camera 3 near-infrared time-series spectroscopic observations of three companions with masses in or near the planetary regime: VHS J125601.92-125723.9 b, GSC 6214-210 B, and ROXs 42 B b. VHS J1256-1257 b exhibits strong total intensity and spectral variability with a brightness difference of 19.3% between 1.1 and 1.7 μm over 8.5 hr and even higher variability at the 24.7% level at 1.27 μm. The light curve of VHS J1256-1257 b continues to rise at the end of the observing sequence so these values represent lower limits on the full variability amplitude at this epoch. This observed variability rivals (and may surpass) the most variable brown dwarf currently known, 2MASS J21392676+0220226. The implied rotation period of VHS J1256-1257 b is ≍21-24 hr assuming sinusoidal modulations, which is unusually long for substellar objects. No significant variability is evident in the light curves of GSC 6214-210 B (<1.2%) and ROXs 42 B b (<15.6%). With a spectral type of L7, an especially red spectrum, and a young age, VHS J1256-1257 b reinforces emerging patterns between high variability amplitude, low surface gravity, and evolutionary phase near the L/T transition.