The problem, pure and simple, is the “stars” in your model. Any time you have a model where five polygons connecting to a single point, you’re going to run into trouble. That trouble will not just be in ZBrush.
In many cases, it’s impossible to avoid those spots entirely. But with good planning you can greatly minimize them and hide the ones you can’t get rid of.
As was recommended, you should look at other models that have really good topology. This will help you learn how to avoid these five-poly intersections.