The pub name "pig and whistle" supposedly comes from "pigging wassail", meaning "good health". Is it true?

Probably not, although it makes a good story. It's just one of many explanations for the name which wasn't anywhere near as common in the past as it is now, as modern pubs compete to come up with names which are different. There is another explanation, favoured by phrases.org. uk, which refers to an old saying from the 1600s when 'pig and whistle' meant 'odds and ends', trivia or to go to rack and ruin. Since pubs were places where much time could be wasted, this might have more truth to it. Full details can be found here - https://www.phrases.org.uk/meanings/pig-and-whistle.html