What a pretty design! I did Chicken Scratch many years ago and it is fun to do. Yes, if you use a hoop for regular embroidery to prevent puckering, you need one for this, too. As to backing--at the time I did this, gingham was 100% cotton. Most manufacturers have added polyester to gingham nowadays and you may see a different result than "back when". I don't know that I'd like the drape having backing on it but, if it's intended for use on a pillow, that might not make a difference. I do think it would be more difficult to pull the needle through though, depending on the size of the square.
Isn't it fun to see how techniques seem to have a come-back as years go by? Don't throw patterns away, even if they have gone out of vogue! Remember candle-wicking? Loved it, too.