To find the length of the edge \( l \) of the square base of the pyramid given its height \( h = 321 \) feet and its slant height \( s = 436.5 \) feet, we can use the relationship between the height, the slant height, and half the base length.
When we draw a vertical line from the apex of the pyramid down to the center of the base, this line (height \( h \)) creates a right triangle with the slant height \( s \) and half the base edge length \( \frac{l}{2} \) as the other two sides.
Using the Pythagorean theorem in this right triangle, we can write the equation:
\[ s^2 = h^2 + \left(\frac{l}{2}\right)^2 \]
Substituting the known values:
\[ (436.5)^2 = (321)^2 + \left(\frac{l}{2}\right)^2 \]
Calculating the squares:
\[ 190,504.25 = 103,041 + \left(\frac{l}{2}\right)^2 \]
Now, subtract \( 103,041 \) from both sides:
\[ 190,504.25 - 103,041 = \left(\frac{l}{2}\right)^2 \]
\[ 87,463.25 = \left(\frac{l}{2}\right)^2 \]
Taking the square root of both sides to solve for \( \frac{l}{2} \):
\[ \frac{l}{2} = \sqrt{87,463.25} \approx 295.7 \]
Now, multiplying by 2 to get the side length \( l \):
\[ l \approx 2 \times 295.7 \approx 591.4 \]
Thus, the length of the edge of the square base, rounded to the nearest tenth, is:
\[ \boxed{591.4 \text{ feet}} \]