Node.js backup cron job
You code looks good and well formatted. There are only a few minor issues:


It seems that you have tried to check it with JSHint but there are some issues that JSHint reports about your code. In particular, semicolons and unused variables.
It is not clear when the character "\n" is required in messages, and when it is not. It looks like sometimes you just forgot it.
This one looks inconsistent. It looks like there is an error:

console.log('The backup file has been saved to your Dropbox: %s', '');

This actually a bad pattern:

if (err) {
    return failWithMySQLError(err);
}


it shortens a code a little but adds impression that the value of the function is important, while it is not.
else is redundant here:

if (err) {
    failWithDropboxError(err);
} else {

and as you see, the long lines complicate the code review.