What methods are appropriate for preparing a 'chow chow' (chayote)?
In my experience, this is most commonly known as chayote; if you're looking for recipes you might have better luck searching for that. (Chow-chow isn't actually on Wikipedia's list of names for it.)

You can generally treat it like any other summer squash. It's a bit firmer and juicier than the usual western summer squashes (yellow squash, zucchini) - maybe somewhere between squash and cucumber in texture - but it has a similarly mild flavor. The skin is also a bit tougher than most summer squash, but it's quite edible, and you may not even notice if it's sliced. So just like other squash, you can slice it and fry or saute it, or boil/steam/bake it whole (and even stuff it). Where you go from there is really up to you; whatever you like to do with summer squash will work fine.

Some people also like to pickle it, or marinate in lime juice; the firmer texture helps it hold up to this.