A brief response to Orin's post below: Perhaps Orin is right, and I'm mistaken; it's hard to tell exactly how these short phrases are likely to be understood.

But my sense is that the "illegal immigrants do the jobs Americans won't do" argument is mostly aimed at quieting people's concerns that illegal immigrants will take jobs that would otherwise go to current American citizens. Don't worry, the argument suggests: These are jobs that wouldn't exist (or would go unfilled) if it weren't for illegal aliens who are willing to do them cheaply.

The more accurate phrasing -- "illegal immigrants just do the jobs Americans won't do for the same low wages that illegal immigrants will take, and it helps our economy to have the jobs done at those low wages" -- carries, I think, quite a different message: It acknowledges that illegal immigrants do take some jobs that would otherwise go to current American citizens (and would go to them at higher wages), but argues (perhaps quite sensibly) that legalizing such immigrants would still be a good idea.

Orin suggests (as I understand it) that the "jobs Americans won't do" argument is basically seen as shorthand for this more precise phrasing. That just doesn't seem to me to be so (as the McCain quotes mentioned in the comments to Orin's post and to my earlier post suggest).