No.
The Blackmagic Intensity is a video capture device... you would be seeing your Xbox display in the preview box of FCP/premier/blackmagic media express... there is also no audio.
Edit: snagged by the above poster, but since I own one, I'll just add that I don't think there is any noticeable lag. I use the Intensity to do a live video feed into Isadora for live theatre projection displays and performances. Less precise than gaming, certainly, but the actors are able to take their cues on time. (we have to use an audio interface and system to run the audio since we're doing this live)