No. Text Animations only work on text.
You'll need to keyframe an animation for the rectangle manually.
As Steve suggests, you would animate the entire Title, using the fixed Effect>Motion>Position (and possibly Motion>Scale). For a different feel, you could even create one Title with the rectangle, and another with the Text, and animate them separately, with perhaps the rectangle entering the frame first, moving rapidly into place, and then the Text following a moment later. Just one of many variations.
Steve also has some good articles on Keyframing on Muvipix. I cannot recall if those are in the free area, or not.