For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $C_n - B_n = A_n^2$?

$\textbf{(A)} \text{ 12} \qquad \textbf{(B)} \text{ 14} \qquad \textbf{(C)} \text{ 16} \qquad \textbf{(D)} \text{ 18} \qquad \textbf{(E)} \text{ 20}$
1. Define the integers \(A_n\), \(B_n\), and \(C_n\) as follows:
   - \(A_n\) is an \(n\)-digit integer where each digit is \(a\).
   - \(B_n\) is an \(n\)-digit integer where each digit is \(b\).
   - \(C_n\) is a \(2n\)-digit integer where each digit is \(c\).

2. Express \(A_n\), \(B_n\), and \(C_n\) in terms of \(a\), \(b\), and \(c\):
   \[
   A_n = a \cdot \frac{10^n - 1}{9}
   \]
   \[
   B_n = b \cdot \frac{10^n - 1}{9}
   \]
   \[
   C_n = c \cdot \frac{10^{2n} - 1}{9}
   \]

3. Given the equation \(C_n - B_n = A_n^2\), substitute the expressions for \(A_n\), \(B_n\), and \(C_n\):
   \[
   c \cdot \frac{10^{2n} - 1}{9} - b \cdot \frac{10^n - 1}{9} = \left(a \cdot \frac{10^n - 1}{9}\right)^2
   \]

4. Simplify the equation:
   \[
   c \cdot \frac{10^{2n} - 1}{9} - b \cdot \frac{10^n - 1}{9} = a^2 \cdot \left(\frac{10^n - 1}{9}\right)^2
   \]

5. Factor out \(\frac{10^n - 1}{9}\) from both sides:
   \[
   c \cdot (10^n + 1) - b = a^2 \cdot \frac{10^n - 1}{9}
   \]

6. Since this is a linear equation in \(10^n\), if it holds for two distinct values of \(n\), it must hold for all \(n\). Therefore, we match coefficients:
   \[
   c = \frac{a^2}{9}
   \]
   \[
   c - b = -\frac{a^2}{9} \implies b = \frac{2a^2}{9}
   \]

7. To maximize \(a + b + c\), we need to maximize \(a\) while ensuring \(b\) and \(c\) are integers. Since \(b\) and \(c\) must be integers, \(a\) must be a multiple of 3.

8. Check the values of \(a\):
   - If \(a = 9\):
     \[
     b = \frac{2 \cdot 9^2}{9} = 18 \quad (\text{exceeds 9, not valid})
     \]
   - If \(a = 6\):
     \[
     b = \frac{2 \cdot 6^2}{9} = 8
     \]
     \[
     c = \frac{6^2}{9} = 4
     \]
     \[
     a + b + c = 6 + 8 + 4 = 18
     \]

Conclusion:
\[
\boxed{18}
\]