A first order filter shifts the phase 45 degrees at the corner frequency, approaching zero as you move into the pass band, and approaching 90 degrees as you move deeper into the rejected frequencies. One way to look at it is that the phase shift is proportional to the slope of the magnitude. (Technically, they are Hilbert transforms of each other.)

A BSC filter is a step filter, where there is a first order pole and a first order zero that are close to each other - less than an octave apart when the boost is less than 6dB. The phase shift of the pole is in the opposite direction from that of the zero, so they partially cancel. The biggest phase shift is at the middle of the transition where the frequency response is steepest, and is about 20 degrees for a 6dB boost. At 4dB (the highest setting of the Fix) it's 14 degrees, and at 2dB boost it's 6.5 degrees. Outside of the transition region there is essentially no phase shift at all, neither above nor below the transition frequency.