1. Use the cross-product condition

Given $\vec d \times \vec b = \vec c \times \vec b$ we get $(\vec d-\vec c)\times \vec b=\vec 0.$

Hence, $\vec d-\vec c$ is parallel to $\vec b$. Therefore, $\vec d=\vec c+\lambda \vec b$ for some scalar $\lambda$.

2. Write $\vec d$ in component form

Given $\vec b=(3,-2,7), \qquad \vec c=(2,-1,4).$ So $\vec d=(2,-1,4)+\lambda(3,-2,7)=(2+3\lambda,-1-2\lambda,4+7\lambda).$

3. Use the dot-product condition

Given $\vec a=(1,4,2), \qquad \vec d\cdot \vec a=24.$ Now, $\vec d\cdot \vec a=(2+3\lambda)(1)+(-1-2\lambda)(4)+(4+7\lambda)(2).$

Compute: $=2+3\lambda-4-8\lambda+8+14\lambda$ $=6+9\lambda.$

Since this equals $24$, $6+9\lambda=24$ $9\lambda=18$ $\lambda=2.$

4. Find $\vec d$

Substitute $\lambda=2$: $\vec d=\vec c+2\vec b=(2,-1,4)+2(3,-2,7)=(2,-1,4)+(6,-4,14)=(8,-5,18).$

5. Compute $|\vec d|^2$

$|\vec d|^2=8^2+(-5)^2+18^2=64+25+324=413.$

6. Check options

The value is $\boxed{413}$ which corresponds to Option B.