Let the co-ordinates of one vertex of \( \triangle ABC \) be \( A(0, 2, \alpha) \) and the other two vertices lie on the line \( \dfrac{x+\alpha}{5} = \dfrac{y-1}{2} = \dfrac{z+4}{3} \). For \( \alpha \in \) \(\mathbb{Z}\) , if the area of \( \triangle ABC \) is \( 21 \) sq. units and the line segment \( BC \) has length \( 2\sqrt{21} \) units, then \( \alpha^{2} \) is equal to ________.
