Can kidney stones cause small crystal pieces to pass in urine?

Yes. Kidney stones are of different sizes. Some are as big as a golf ball or more and some are minute crystals like sand or gravel. Also, some are very small and not even noticeable. Many have crystals in urine that are visible only under the microscope.
Yes. Kidney stones are small crystals from dissolved material in the urine that crystallize in the urinary outflow tract. If they can pass, you are lucky. And you might see them in the urine. If this is a regular occurrence, you might want to save a few, give them to your family doctor to have them analyzed to get suggestions as to how to avoid their formation.