UltimateTicTacToe - ActionScript Style!
Overall this is looking pretty. I do have a few questions and suggestions based on what I see.


Constructor
Your class extends MovieClip therefore it IS a Movieclip. What is the purpose of the inner _square class variable? Is SquareSpace meant to be purely a logic controller class with _square being the visual representation of the square? If so there is no need for SquareSpace to extend MovieClip. I would look at EventDispatcher or just plain Object.
addSquare/sendClick
I'm not sure I'm understanding the purpose of these methods. Can you explain your thought behind them?
changeSquareState
For state control/detection you generally want to use a switch statement. That way adding more states in the future is much easier.
The State Constants
This is more of a cosmetic comment but I usually prepend any constants with the word STATE_ so that you/other developers can easily tell that those are values intended to be used for state control and passed into changeSquareState.
Argument Error
A really minor comment but I like to include the state which is not acceptable in the error statement. Makes debugging in the future easier. Not a big deal though.
Future Design
As you build out this game do you plan on using the Flash timeline to position and configure your assets or are you going pure AS3?
Package
Generally you want to organize each of your classes into a package. If you're not using a proper IDE like FlashBuilder, FlashDevelop, or FDT this will be a bit of a pain but it's good practice none the less. Packages generally look like reversed domain names so for example your SquareSpace class might be in the following package com.alex.ultimatetictactoe.square