Yes, you can override functions. But the example you gave is not overriding - it's overloading. For overriding a function, you basically just create a function with the same signature and everything in a subclass and add the word "override" right before it.
You can't directly overload a function though. If you want a variable number of parameters, you have to use optional parameters instead. Like this:
function init(str:String = null):String
{
if (str == null)
{
// do one thing
return null;
}
else
{
// do another thing
return "someString";
}
}
And that's about the best you're going to be able to do in AS3. The inability to overload functions, at least strictly speaking, is a fairly common complaint and obvious shortcoming of the language.