Ed and Sue bike at equal and constant rates.  Similarly, they jog at equal and constant rates, and they swim at equal and constant rates.  Ed covers $ 74$ kilometers after biking for $ 2$ hours, jogging for $ 3$ hours, and swimming for $ 4$ hours, while Sue covers $ 91$ kilometers after jogging for $ 2$ hours, swimming for $ 3$ hours, and biking for $ 4$ hours.  Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour.  Find the sum of the squares of Ed's biking, jogging, and swimming rates.
1. Let's denote Ed's and Sue's biking, jogging, and swimming rates by \( b \), \( j \), and \( s \) respectively. According to the problem, we have the following equations based on their activities:
   \[
   2b + 3j + 4s = 74 \quad \text{(Equation 1)}
   \]
   \[
   4b + 2j + 3s = 91 \quad \text{(Equation 2)}
   \]

2. To eliminate one of the variables, we subtract Equation 1 from Equation 2:
   \[
   (4b + 2j + 3s) - (2b + 3j + 4s) = 91 - 74
   \]
   Simplifying this, we get:
   \[
   2b - j - s = 17 \quad \text{(Equation 3)}
   \]

3. Next, we multiply Equation 3 by 3 to facilitate further elimination:
   \[
   3(2b - j - s) = 3 \cdot 17
   \]
   \[
   6b - 3j - 3s = 51 \quad \text{(Equation 4)}
   \]

4. We add Equation 4 to Equation 1:
   \[
   (6b - 3j - 3s) + (2b + 3j + 4s) = 51 + 74
   \]
   Simplifying this, we get:
   \[
   8b + s = 125 \quad \text{(Equation 5)}
   \]

5. We also add Equation 4 to Equation 2:
   \[
   (6b - 3j - 3s) + (4b + 2j + 3s) = 51 + 91
   \]
   Simplifying this, we get:
   \[
   10b - j = 142 \quad \text{(Equation 6)}
   \]

6. From Equation 5, we solve for \( s \):
   \[
   s = 125 - 8b
   \]

7. From Equation 6, we solve for \( j \):
   \[
   j = 10b - 142
   \]

8. We substitute \( s \) and \( j \) back into Equation 3 to find \( b \):
   \[
   2b - (10b - 142) - (125 - 8b) = 17
   \]
   Simplifying this, we get:
   \[
   2b - 10b + 142 - 125 + 8b = 17
   \]
   \[
   0b + 17 = 17
   \]
   This confirms that our values for \( b \), \( j \), and \( s \) are consistent.

9. We now solve for \( b \) using the bounds derived from Equations 5 and 6:
   \[
   125 \geq 8b \implies b \leq 15.625
   \]
   \[
   142 \leq 10b \implies b \geq 14.2
   \]
   Since \( b \) must be a whole number, we conclude:
   \[
   b = 15
   \]

10. Substituting \( b = 15 \) back into the expressions for \( s \) and \( j \):
    \[
    s = 125 - 8 \cdot 15 = 125 - 120 = 5
    \]
    \[
    j = 10 \cdot 15 - 142 = 150 - 142 = 8
    \]

11. Finally, we calculate the sum of the squares of Ed's biking, jogging, and swimming rates:
    \[
    15^2 + 5^2 + 8^2 = 225 + 25 + 64 = 314
    \]

The final answer is \(\boxed{314}\).