Note that $12^2=144$ ends in two $4$s and $38^2=1444$ end in three $4$s. Determine the length of the longest string of equal nonzero digits in which the square of an integer can end.
1. **Check for squares ending in two identical digits:**
   - We need to check if a square can end in two identical digits, specifically for digits 1 through 9.
   - Consider the last two digits of a number \( n \) and its square \( n^2 \). We can use modular arithmetic to simplify this check.

   For example, let's check if a square can end in two 1s:
   \[
   n^2 \equiv 11 \pmod{100}
   \]
   We need to find \( n \) such that \( n^2 \equiv 11 \pmod{100} \). By testing all numbers from 0 to 99, we find that no such \( n \) exists. Similarly, we can check for other digits.

   - For digits 2, 3, 5, 6, 7, 8, and 9, we can use the same method to show that no square ends in two identical digits.

2. **Check for squares ending in three identical digits:**
   - We need to check if a square can end in three identical digits, specifically for digits 1 through 9.
   - Consider the last three digits of a number \( n \) and its square \( n^2 \). We can use modular arithmetic to simplify this check.

   For example, let's check if a square can end in three 4s:
   \[
   n^2 \equiv 444 \pmod{1000}
   \]
   We need to find \( n \) such that \( n^2 \equiv 444 \pmod{1000} \). By testing all numbers from 0 to 999, we find that no such \( n \) exists. Similarly, we can check for other digits.

3. **Check for squares ending in four identical digits:**
   - We need to check if a square can end in four identical digits, specifically for digits 1 through 9.
   - Consider the last four digits of a number \( n \) and its square \( n^2 \). We can use modular arithmetic to simplify this check.

   For example, let's check if a square can end in four 4s:
   \[
   n^2 \equiv 4444 \pmod{10000}
   \]
   We need to find \( n \) such that \( n^2 \equiv 4444 \pmod{10000} \). By testing all numbers from 0 to 9999, we find that no such \( n \) exists. Similarly, we can check for other digits.

4. **Conclusion:**
   - From the above checks, we can conclude that the longest string of equal nonzero digits in which the square of an integer can end is three.

The final answer is \( \boxed{ 3 } \).