Add the potatoes to a large pot of cold water and bring to a boil. Boil for 30-35 minutes or until easily pierced with a fork. Drain and set aside.
(If you have not yet prepared the hard boiled eggs) In a small pot of cold water, add eggs and bring to a boil. Boil for 14 minutes then drain, rinse with cool water, and set aside. Peel and chop once cool.
Peel the potatoes and chop into approximately ½" pieces. Transfer to a mixing bowl and sprinkle with vinegar, tossing in the vinegar and setting aside to cool completely.
Add celery, green onions, and eggs to the potato mixture.
In a separate bowl mix mayonnaise, mustard, and seasonings. Fold into the potato mixture.
Chill at least 3 hours, preferably overnight before serving.