A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.

[asy]
import three; import solids;
size(5cm);
currentprojection=orthographic(1,-1/6,1/6);

draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,360)),white,nolight);

triple A =(8*sqrt(6)/3,0,8*sqrt(3)/3), B = (-4*sqrt(6)/3,4*sqrt(2),8*sqrt(3)/3), C = (-4*sqrt(6)/3,-4*sqrt(2),8*sqrt(3)/3), X = (0,0,-2*sqrt(2));

draw(X--X+A--X+A+B--X+A+B+C);
draw(X--X+B--X+A+B);
draw(X--X+C--X+A+C--X+A+B+C);
draw(X+A--X+A+C);
draw(X+C--X+C+B--X+A+B+C,linetype("2 4"));
draw(X+B--X+C+B,linetype("2 4"));

draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,240)),white,nolight);
draw((-2,-2*sqrt(3),0)..(4,0,0)..(-2,2*sqrt(3),0));
draw((-4*cos(atan(5)),-4*sin(atan(5)),0)--(-4*cos(atan(5)),-4*sin(atan(5)),-10)..(4,0,-10)..(4*cos(atan(5)),4*sin(atan(5)),-10)--(4*cos(atan(5)),4*sin(atan(5)),0));
draw((-2,-2*sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype("2 4"));
[/asy]
1. **Determine the volume of the cylindrical barrel:**
   The volume \( V_{\text{cylinder}} \) of a cylinder is given by the formula:
   \[
   V_{\text{cylinder}} = \pi r^2 h
   \]
   where \( r \) is the radius and \( h \) is the height. For this problem, \( r = 4 \) feet and \( h = 10 \) feet.
   \[
   V_{\text{cylinder}} = \pi (4)^2 (10) = 160\pi \text{ cubic feet}
   \]

2. **Determine the volume of the cube:**
   The volume \( V_{\text{cube}} \) of a cube is given by the formula:
   \[
   V_{\text{cube}} = s^3
   \]
   where \( s \) is the side length. For this problem, \( s = 8 \) feet.
   \[
   V_{\text{cube}} = 8^3 = 512 \text{ cubic feet}
   \]

3. **Determine the volume of water displaced by the cube:**
   When the cube is placed in the barrel with its diagonal vertical, the entire volume of the cube displaces an equivalent volume of water. Therefore, the volume of water displaced \( V_{\text{displaced}} \) is equal to the volume of the cube.
   \[
   V_{\text{displaced}} = 512 \text{ cubic feet}
   \]

4. **Calculate \( v^2 \):**
   Given that \( v \) is the volume of water displaced, we have \( v = 512 \). We need to find \( v^2 \).
   \[
   v^2 = (512)^2
   \]
   To calculate \( 512^2 \):
   \[
   512^2 = (500 + 12)^2 = 500^2 + 2 \cdot 500 \cdot 12 + 12^2 = 250000 + 12000 + 144 = 262144
   \]

The final answer is \( \boxed{262144} \).