Point B lies on line segment \overline{A C} with A B=16 and B C=4. Points D and E lie on the same side of line A C forming equilateral triangles \triangle A B D and \triangle B C E. Let M be the midpoint of \overline{A E}, and N be the midpoint of \overline{C D}. The area of \triangle B M N is x. Find x^{2}.