The short answer is yes, the long answer is that it depends. In order to understand which spirits can be used it's helpful to first understand how the colour changing effect works. The blue pea flowers in the kit act as an indicator of acidity or alkalinity (just like the indicator you used to use in Chemistry at school). When they're infused in a neutral or alkaline liquid the colour is blue. When the liquid is acidic then the colour is pink. Gin has an almost neutral ph, whereas tonic water or lemon juice is much more acidic, hence adding the tonic water to the gin causes the cocktail to magically turn from blue to pink.
Whether you can use other spirits depends on two things. Most obviously is the colour of the spirit, if you're using something which is already coloured then the effect may not be that noticeable. Whether you can use another spirit also depends on the acidity of the spirit. If the spirit itself is acidic then there would be no colour change when it is mixed with an acidic mixer.
You can see from the table below, which was produced by the Stirred, not Shaken blog, that there is considerable variation in the acidity of different spirits. Most gins are at the neutral end of the spectrum, whereas other liquors can be very acidic.