I am trying to figure out the same question. Crossword Compiler has two web export functions under "File"-->"Web Export". The first, "Publish to the web...", sends the puzzle into some kind of database page. The applet can apparently be run from there but I don't think you can move it out to a different site. The second option under the same menu, "Export files..." gave me slightly better results, in that it generated the html files to upload to a website of my choosing.
At this point though, I'm stuck--I tried Facebook but couldn't find a way of uploading html files (I don't have MySpace so I don't know if it's the same case over there). There must be a way to upload them on a independent personal blog as opposed to a social networking site, e.g. via Typepad (used by B.E. Quigley) or Blogger, but I haven't looked into it further. If anybody has further advice on this part please do share. I'm also interested to know if anyone has successfully used Facebook as a platform for interactive crosswords; it sounds technically feasible but maybe I just didn't look in the right place.