You encounter Yuffie by finding her in a random encounter where she will appear as 'Mystery Ninja'. You find her in the forests outside of Junon. After beating her, you'll appear in a unique area. DO NOT APPROACH THE SAVE POINT. Walk up to her and talk to her. When prompted, here are the 'correct' answers:
"Not interested."
"...petrified."
"Wait a second!"
"That's right."
"Let's hurry on."
If you pick these answers you will enter the name select screen and she will then join your party.