Convert Ternary Expression to a Binary Tree
You can delete the BST class. You don't use it anywhere.
Error handling looks weird to me. I suggest throwing an exception if the input is invalid. You also don't check all possible cases. For instance, you code prints a tree ? | a | b | for the input ??a:b, which is clearly invalid. I'd recommend either adding proper error handling or just dropping it altogether (the problem statement says that the string is a ternary expression, anyway). 
Comments should not repeat the code. If you have something like // if it is a valid character, it's a good indicator that the following check should be moved to a separate method with a proper name (something like private boolean isValidCharacter(Node node, String expression, int index).