Answer: 224

Small Hint:

The three circle centers are collinear because their pairwise distances are $6,$ $3,$ and $9$

Big Hint:

Use signed distances from the two smaller centers to the common tangent to find its distance from the large center

Solution:

Put the radius-$9$ circle at the origin and the smaller centers at $(-6,0)$ and $(3,0).$ Write the common external tangent as $\mathbf n\cdot(x,y)=c,$ where $\mathbf n$ is a unit normal. Its signed distances from the two centers differ by $6-3=3,$ so $9n_x=3$ and $n_x=\frac{1}{3}.$ Using the distance $3$ from $(-6,0)$ gives $|c|=5.$ Thus the chord lies $5$ units from the large center, and its squared length is $$4(9^2-5^2)=4\cdot56=224.$$