OO design for Tic Tac Toe program
Overall this looks pretty good. Just a couple of things come to mind:

Square.js


The setSymbol method should throw errors instead of returning strings, especially since I don't see any place where those strings are handled in a manor allowing the user to correct their mistake.
The error message when the square is occupied just needs a slight grammatical correction: This square is already filled (filled instead of filler)


TTTGame.js


Create all methods on the prototype. You aren't using any private function scope variables, so putting the methods on the prototype will be more efficient.


General Comments

The name TTTGame is kind of obtuse. I know it is probably an abbreviation for "Tick Tack Toe Game" but I think a more descriptive name is in order. All around, throwing these classes into a namespace would be useful. Consider the semantic differences between these two sets of classes:


Square (is it a tick tack toe square or a two dimensional representation of a plane in mathematics)
Board (Diving board? Game board? Discussion board?)
TTTGame (What does TTT stand for?)


Vs:


TickTackToe.Square
TickTackToe.Board
TickTackToe.Game


No explanation necessary for the second set of class names.

All in all, this looks pretty good.