@CompletedDate and @CompletedDate2 should be of data type date, no other data type. This means that in SQL Server the question about format is irrelevant. Date does not have a format, it is just a value.
On the other hand, on the client where you actaully pass the data and get it from the users, you need to consider this. Then again, this is often automatic. If you use a date-picker, I guess it outputs some sort of date object, which you pass to your parameter. But even if it is a string, you just take the string and let the regional settings on the machine decide, so that users can use whichever format they like.