I would like to lock focus on a face or an eye, and have the camera keep re-focusing in the middle of a burst as the subject moves.
Can any of the current Rebel series continue re-focusing in the middle of a burst, or is this feature reserved for the 50D or 7D (or higher)?
Yes, all dslrs can do that, including entry-level dslrs from Canon. To my knowledge Canon calls this AF mode `AI Servo' while Nikon calls it AF-C (= continuous AF, a more intuitive name, but anywho).
One word of caution: even though you have this ability built into your camera, to master it is quite difficult. Since I don't use Canons, I don't know all the different settings (I'm sure they exist in one form or another), but on my Nikon, you have 3 x 3 different AF modes, on the bigger models it's at least 4 x 3: to know which mode you want to use requires some practice.
On the other hand, with kit lenses, you will have less focus issues as the depth of field is comparatively large. It's much more important with more expensive lenses, e. g. primes with large apertures or tele lenses. With my bazooka zoom, I notice if the focus point is just a few mm off (at a distance of, say, 2-3 m).
BTW, I heard that Nikon was somehow better at predicting movement and maintaing focus on a moving subject, but my friends with whom I share lenses shoot Canon, so that's what I'm using. Any truth to Nikon having superior focusing ability on any of their crop sensor cameras?
There is anecdotal evidence that Nikon tunes its autofocus systems to value accuracy over speed. I've seen this in several head-to-head tests, but even if there is a difference, I don't think it matters as much with today's cameras.
Currently, I can think of only one Canon camera that is at a significant disadvantage when it comes to the AF system and that's the 5D Mark II: it essentially retains the fairly ancient AF system of the 5D and even the Canon 40D/50D feature a more beefy AF system than the 5D Mark II! Since you're not interested in this camera, I don't think it's particularly relevant.
Other than that, I think neither company has a significant edge when it comes to AF systems. In any case, not significant enough to switch from C to N or vice versa.