You'd get very little juice doing that and you'd be pressing against a strainer quite a bit. It's a waste. Trust me, we use our Vitapreps/Vitamix for a whole lot of things (they're the favorite kitchen tool of a lot of chefs), we'll push it, but they do not get used to juice because they're not meant for that and don't do a very good job.
The nearby health food store uses a juicer and a Vitamix to make smoothies that contain vegetables.