In your project access the Tools menu and select Customize.
Select Project Entities.
Expand the tree item for Defect.
Click on User Fields.
Click the toolbar button New Field.
In the Label field enter the text you want displayed on the screen as the label for the new field.
In the Type field select Date.
Save the new field.
To make the field required in only certain circumstances will require you to customize the workflow. To place the field in a specific place within the fields displayed also requires workflow customization. For more information on that, and on customizing projects in general, refer to the Administrator Guide. There is a whole section on Workflow Customization that includes lots of examples.
(Opinions and information contained in this post are wholly my own and do not reflect the opinions of my employer.)