To determine whether there exists an infinite sequence of nonzero digits $a_1, a_2, a_3, \ldots$ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1} \cdots a_1}$ is a perfect square, we analyze the structure of perfect squares and the requirements of the sequence.

1. Understanding the Problem:
The problem asks for an infinite sequence of nonzero digits, such that the number formed by the first $k$ digits in reverse order is a perfect square for $k > N$.

2. Properties of Perfect Squares:
- A perfect square $b^2$ (for some integer $b$) typically has a number of digits that increases roughly by a factor of 2 for each additional digit in $b$.
- The structure and distribution of digits in perfect squares follow particular patterns. For instance, the last digit of a perfect square ends only in 0, 1, 4, 5, 6, or 9.

3. Contradiction via Limitations of Nonzero Digits:
- The sequence $\overline{a_k a_{k-1} \cdots a_1}$, composed entirely of nonzero digits, implies the number does not end in zero.
- As $k \to \infty$, the sequence length $k$ should still form a perfect square. Each perfect square needs to adhere to integer properties such as divisibility and congruence relations (e.g., a number conservatively ending in certain digits, discussed before).

4. Logical Analysis:
- Suppose for contradiction that such a sequence and $N$ exist. For very large $k$, the number of digits in a perfect square must align with $\lfloor \log_{10}(b^2) \rfloor + 1$, where $b^2 = \overline{a_k a_{k-1} \cdots a_1}$.
- Consider ever-increasing $b$, and hence $b^2$, to maintain the perfect square property.
- However, the requirement for all digits $a_1, a_2, \ldots, a_k$ to be nonzero severely restricts the possibility for all $b^2$'s necessary divisibility and ending digit patterns, especially as $b$ becomes very large (i.e., imbalances the density of typical nonzero digit ends).

5. Conclusion:
- The structural constraints and requirements imposed on $b^2$ by the infinite sequence of nonzero digits lead to an eventual impossibility.
- There can't be an infinite sequence where every freshly formed $\overline{a_k a_{k-1} \cdots a_1}$ remains a perfect square past a certain point $N$.

Thus, there exists no such infinite sequence satisfying the problem's conditions. The answer is:
$$\boxed{\text{No}}$$