It took me a couple of days but I found it out. So for future reference: here is my solution:
I extended the DateChooser and added an override on the function updateDisplayList(w:Number, h:Number) (In this function the SMTWTFS day names have been set).
In the updateDisplayList you can get the mx_internal::dateGrid.dayBlockArrays[column][row] Containing all the values for the CalendarLayout. In that array / array the first row on each column is the one of the SMTWTFS days. The other rows are the dayNumbers. Once I found that out it is a matter of determining what a weekend day was and adjust the colors accordingly. As I show below:
override protected function updateDisplayList(w:Number, h:Number):void
    {
        super.updateDisplayList(w, h);
        // Now the dayBlocksArray has been filled. The Array looks as follows 
        // dayBlocksArray["rows"]["columns] .... therefor [i][0 (zero)] will always get the dayNames (SMTWTFS)
        // Compare the dayNames with the set this.dayNames (which always start with sunday) and find the weekend days
        var colIndex:uint = 0;
        var rowIndex:uint = 1; // The first row (SMTWTFS) is handled seperately
        var currentColumn:Array;
        var dayName:UITextField;
        var backgroundColor:uint = this.getStyle("dayNamesBackgroundColor");
        var isWeekendCol:Boolean = false;
        var currentTextFormat:TextFormat;
        // When the style is not found the default of white will be used.
        if (!backgroundColor)
        {
            backgroundColor = 0xFFFFFF;
        }
        for (colIndex; colIndex < 7; colIndex++)
        {
            // First determine if the first item in this row (SMTWTFS) is a week/weekend day
            currentColumn = mx_internal::dateGrid.dayBlocksArray[colIndex];
            dayName = currentColumn[0];
            // Determine if this is a weekend row
            // The dayNames array is fixed in the order of index 0 = sunday and index 6 = saturday. 
            // Therefor check of the text value of the current dayName is equal to either of 
            // those two. If it is we are dealing with a weekend column
            isWeekendCol = dayName.text == this.dayNames[0] || dayName.text == this.dayNames[6];
            if (isWeekendCol)
            {
                // Set the color
                currentTextFormat = dayName.getTextFormat();
                currentTextFormat.color = getStyle("weekendHeaderColor");
                dayName.setTextFormat(currentTextFormat);
                // Set the background color
                dayName.background = true;
                dayName.backgroundColor = backgroundColor;
            }
            else
            {
                currentTextFormat = dayName.getTextFormat();
                currentTextFormat.color = getStyle("weekHeaderColor");
                dayName.setTextFormat(currentTextFormat);
                // Set the background color
                dayName.background = true;
                dayName.backgroundColor = backgroundColor;
            }
            // Reset the rowIndex
            rowIndex = 1;
            // Now go through all the other rows of this column
            for (rowIndex; rowIndex < currentColumn.length; rowIndex++)
            {
                dayName = currentColumn[rowIndex];
                if (isWeekendCol)
                {
                    dayName.setColor(getStyle("weekendColor"));
                }
                else
                {
                    dayName.setColor(getStyle("weekColor"));
                }
            }
        } 
}
In the CSS file I added the following styles:
DateChooser {
cornerRadius: 0;
   headerColors: #FFFFFF, #FFFFFF;
   todayColor: #00448c;
   border-style:none;
   dropShadowEnabled: false;
   fontFamily: myGeorgia;
   dayNamesBackgroundColor: #ECECEC;
   weekHeaderColor:#444444;
   weekendHeaderColor:#DDDDDD;
   weekColor:#00317F;
   weekendColor: #DDDDDD;
   headerStyleName: "dateChooserHeaderStyle";
   comboBoxStyleName: "comboBoxStyleName";
}
The most interesting styles here are the custom style "dayNamesBackgroundColor" (which is used to give a background color to the SMTWTFS set) and the custom styles "weekendHeaderColor", "weekHeaderColor", "weekColor", "weekendColor"
I read these colors in the method above to get full control for the difference in week/weekend colors where the SMTWTFS set could get different colors than the day number
Hope this will help other people in the future. Took me a lot of time to figure it out :)