When you "reload" your collection you basically replace all values in it with new values. Even those that look and feel identical are in fact new items. So how do you want to reference the same item in the list when it is gone? You could certainly use a hack where you determine the item that was selected by its properties and reselect it (i.e. do a LINQ search through the list and return the ID of the matching item, then reselect it). But that would certainly not be using best practices.
You should really only update your collection, that is remove invalid entried and add new entries. If you have a view connected to your collection all the sorting and selecting and whatnot will be done automagically behind the scenes again.
Edit:
var tmp = this.listBox1.SelectedValue;
this._customers.Clear();
this._customers.Add(item1); this._customers.Add(item2);
this._customers.Add(item3); this._customers.Add(item4);
this.listBox1.SelectedValue = tmp;
in the method that does the reset/clear works for me. I.e. that is the code I put into the event handling method called when pressing the refresh button in my sample app. That way you dont even need to keep references to the customer objects as long as you make sure that whatever your ID is is consistent. Other things I have tried, like overwriting the collections ´ClearItems()´ method and overwriting ´Equals()´ and ´GetHashCode()´ didn't work - as I expected.