A word of warning: be careful when stereotyping Requirements - the relationship between the element stereotype and the requirement types configured into the project (Settings - Project Types) is a bit murky.
That said, there is no way to modify EA's dialogs. What you can do is create your own property dialog and have EA open that when the user double-clicks the element.
In order to achieve this, you need to create an Add-In which catches the Context Item Event EA_OnContextItemDoubleClicked. In your event handler, return true to inform EA that you're handling the event (which prevents EA from opening the default property dialog), and open your own custom dialog.
The same event is fired when the user hits Enter with the element selected. The default property dialog can still be opened by pressing Alt-Enter.
As an alternative, you might want to look into the third-party extension eaForms, which allows you to create your own custom property dialogs without writing your own Add-In.