If you use contractions like don\'t inside of a string then you need to escape the apostrophe if you\'re using single quotes, which is annoying and can make the string a bit harder for humans to parse. I like the aesthetics of double-quotes better too, though that\'s not a compelling reason.
For such cases I find the Python style most appropriate. It says that you use some consistent default (e.g. always single quote), but for all a strings that require escaping, you should use the delimiter with the least (preferably no) escaping.
Note that this is easier in Python than JavaScript because Python provides more string delimiters:
'He said hello.'
'He said "hello".'
"Don't say hello.'
'''Don't say "hello".'''
r'Some regex\sstuff'
My toy programming language uses “ and ” for string literals. It counts nesting, so you can write “Don’t say “Hello””. There is no escaping. In fact, you can quote any piece of code by simply enclosing it in “ and ”. Code is commented out by turning it into an unused string literal.
I can't personally imagine this solution would be any less confusing, given that the new preferred apostrophe character according to this source is specifically a right single quote character.
