In a word... no. The use of a click track during the recording process is entirely optional and should be decided in terms of what best serves the performance of each song. Some songs may benefit from a metronomic feel, while others will need the natural push and pull of a band playing 'free time'.
Another thing to bear in mind is that some drummers are entirely comfortable playing with a click while others have never done it before. The recording studio is definitely not the place to learn this skill but it is a very useful ability to have, so I would encourage all drummers to practice playing to a metronome at least some of the time.
It's worth discussing this and other aspects of the session beforehand, so feel free to call or email and we'll be happy to talk about the logistics of the recording process and make recommendations as to the best approach for your band.