Is feature importance from classification a good way to select features for clustering?
It might make sense, but it depends what you're trying to do:

If the goal is to predict the binary target for any instance, a classifier will perform much better.
If the goal is to group instances by their similarity, loosely taking the binary target into account indirectly, then clustering in this way makes sense. This would correspond to a more exploratory task where the goal is to discover patterns in the data, focusing on the features which are good indicators of the target (it depends how good they actually are).