I need to find a specific set of users for a particular period and then see if they've returned in another date range, how can I create the segment?
That's because you are using Date Range in the segment. You can just use the segment with the condition in a different time period - 23 June - 9 July.
You will be able to see the Return Frequency report with line items:
Apart from this, it isn't possible to customize the visitor from a particular time frame, like 01 June - 18 June.
Hope it clarifies!