The most likely reason for use of single vs. double in different libraries is programmer preference and/or API consistency. Other than being consistent, use whichever best suits the string.

Using the other type of quote as a literal:

alert('Say "Hello"');
alert("Say 'Hello'");

This can get complicated:

alert("It's \"game\" time.");
alert('It\'s "game" time.');

Another option, new in ECMAScript 6, is template literals which use the backtick character:

alert(`Use "double" and 'single' quotes in the same string`);
alert(`Escape the \` back-tick character and the \${ dollar-brace sequence in a string`);

Template literals offer a clean syntax for: variable interpolation, multi-line strings, and more.

Note that JSON is formally specified to use double quotes, which may be worth considering depending on system requirements.

Do:

var isTrueSet = (myValue === 'true');

using the identity operator (===), which doesn't make any implicit type conversions when the compared variables have different types.

Don't:

You should probably be cautious about using these two methods for your specific needs:

var myBool = Boolean("false");  // == true

var myBool = !!"false";  // == true

Any string which isn't the empty string will evaluate to true by using them. Although they're the cleanest methods I can think of concerning to boolean conversion, I think they're not what you're looking for.