Preheat the oven to 350 Degrees F. Grease an 8-inch square baking pan and set aside. In a medium mixing bowl, beat the eggs and butter together until fluffy. Beat in the eggs and vanilla, blending well. Add the flour and beat until smooth. Pour half of the batter into the prepared pan and set aside. Melt the chocolate chips over very low heat. Add the melted chips to the remaining batter, blending well. Spread the chocolate batter over the white batter in the pan. Zigzag, with a knife or spatula, through the layered batters to marble the chocolate into the white. Bake for 25 minutes. Place the pan on a wire rack to cool until cold to the touch, before cutting into squares.