Should I perform customer segmentation before performing churn prediction?
There is no direct reason to segment your customers before estimating their churn probability, unless you know of business reasons for the segments to behave in a substantially different way with regard to churn, or the business users of the model for the different segments will have differing requirements (e.g. on model interpretability). Otherwise, there is nothing that a well-trained churn model could not do that two models (clustering and churn classification) would.
You can actually see for yourself if the approach brings any benefit - add the segmenting features to the churn model input and check whether they have important influence on its outcomes (they are high in the decision tree, have high feature importance, etc.) If they do, segmenting the customers and training separate churn models might bring benefits, especially if you have strict requirements on model interpretability.