For how many integers $n$, does the equation system \[\begin{array}{rcl}
2x+3y &=& 7\\
5x + ny &=& n^2
\end{array}\] have a solution over integers?

$ 
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 8
\qquad\textbf{(E)}\ \text{None of the preceding}
$
To determine the number of integer values of \( n \) for which the system of equations
\[
\begin{array}{rcl}
2x + 3y &=& 7 \\
5x + ny &=& n^2
\end{array}
\]
has integer solutions, we will analyze the system step-by-step.

1. **Express \( x \) in terms of \( y \) from the first equation:**
   \[
   2x + 3y = 7 \implies 2x = 7 - 3y \implies x = \frac{7 - 3y}{2}
   \]
   For \( x \) to be an integer, \( 7 - 3y \) must be even. Since 7 is odd, \( 3y \) must also be odd, implying \( y \) must be odd.

2. **Substitute \( x \) into the second equation:**
   \[
   5x + ny = n^2 \implies 5 \left( \frac{7 - 3y}{2} \right) + ny = n^2
   \]
   Simplify the equation:
   \[
   \frac{35 - 15y}{2} + ny = n^2
   \]
   Multiply through by 2 to clear the fraction:
   \[
   35 - 15y + 2ny = 2n^2
   \]
   Rearrange the equation:
   \[
   2n^2 - 2ny + 15y = 35
   \]
   Factor out common terms:
   \[
   2n^2 - 2ny + 15y = 35 \implies 2n^2 + y(15 - 2n) = 35
   \]

3. **Analyze the quadratic form:**
   We need to find integer solutions for \( n \) such that the equation holds. We can rewrite the equation as:
   \[
   2n^2 + y(15 - 2n) = 35
   \]
   Rearrange to isolate \( y \):
   \[
   y(15 - 2n) = 35 - 2n^2
   \]
   Since \( y \) must be an integer, \( 15 - 2n \) must divide \( 35 - 2n^2 \).

4. **Factorization approach:**
   We can use Simon's Favorite Factoring Trick (SFFT) to factorize:
   \[
   2n^2 + y(15 - 2n) = 35 \implies (2n - 15)(2n - 15) = 155
   \]
   The factors of 155 are \( \pm 1, \pm 5, \pm 31, \pm 155 \).

5. **Determine possible values of \( n \):**
   For each factor pair \((a, b)\) such that \( ab = 155 \), we solve:
   \[
   2n - 15 = a \quad \text{and} \quad 2n - 15 = b
   \]
   Solving for \( n \):
   \[
   2n = a + 15 \implies n = \frac{a + 15}{2}
   \]
   We check each factor pair:
   \[
   \begin{array}{rcl}
   a = 1, & b = 155 & \implies n = \frac{1 + 15}{2} = 8 \\
   a = -1, & b = -155 & \implies n = \frac{-1 + 15}{2} = 7 \\
   a = 5, & b = 31 & \implies n = \frac{5 + 15}{2} = 10 \\
   a = -5, & b = -31 & \implies n = \frac{-5 + 15}{2} = 5 \\
   a = 31, & b = 5 & \implies n = \frac{31 + 15}{2} = 23 \\
   a = -31, & b = -5 & \implies n = \frac{-31 + 15}{2} = -8 \\
   a = 155, & b = 1 & \implies n = \frac{155 + 15}{2} = 85 \\
   a = -155, & b = -1 & \implies n = \frac{-155 + 15}{2} = -70 \\
   \end{array}
   \]

6. **Count the distinct integer values of \( n \):**
   The distinct values of \( n \) are \( -70, -8, 5, 7, 8, 10, 23, 85 \), which gives us a total of 8 values.

The final answer is \(\boxed{8}\)