gistfile1.txt

function MyObject () {
    // do stuff
}

// now to export this as a nodejs module do
module.exports = MyObject;

// =====

// in another file (in the same folder) you can now do
var MyObject = require("./MyObject");
// use it just like before
var obj = new MyObject();

In Summary, here's the changes I needed to make:

1. At the top of the class TimeEntry.js, added a require line for the dependency:

var FreshBooks_Element = require('./Element');

2. After the declaration of the function constructor in TimeEntry.js:
   function FreshBooks_TimeEntry()
   {
   this.elementName = "time_entry";
   ...
   }
   Added, per http://nodejs.org/api/modules.html#module.exports and http://stackoverflow.com/questions/8296379/create-a-class-in-nodejs:
   module.exports = FreshBooks_TimeEntry;

3. After the declaration of the function constructor in Element.js:

function FreshBooks_Element()
{
...
}

Added:
module.exports = FreshBooks_Element;