When you simply paste text in Illustrator, it creates a text object. A text object can't be force justified in Illustrator. You must first create a text frame by drawing a box shape with the text tool. Click into the frame and paste in your text, then you can control justification.
[Added further explanation]
Illustrator allows you to use text in a number of ways. By default, when you paste text from the clipboard with nothing selected, Illustrator just drops the copied text as a basic text object. Text objects are good for single words or small phrases, but not for body copy because they have no defined boundaries. For that, you have to select the text tool from the tools panel and "draw" a text frame. With the text tool selected, click and drag a box from the top left to the lower right to create a new text frame. (Newer versions of Illustrator automatically fill the frame with placeholder text. Older versions just leave the frame empty.) Select the placeholder text (if necessary) and paste your copied text into the frame. You can drag the handles of the text frame to change its dimensions to suit your needs. Once the text is inside the frame, you can use the justification options located on the Paragraph panel to select your desired justification method.