Why can an integer written $2$ times in a row never be a perfect square? It seems to be true for all natural numbers below $1,000,000$.
I am really stuck any kind of help will be appreciated!
This is not true in general.

Here is a counterexample: 4049586776940495867769 is the square of 63636363637

Why did I have to come up with such a huge counterexample? The reason is that the "conjecture" indeed works for small numbers. Here is why : Consider $\overline{NN}$ where $N$ is a $d$-digit number so that $\overline{NN}$ has $2d$ digits. $\overline{NN} = N \times (10^d+1)$. If $10^d+1$ is squarefree, $\overline{NN}$ has to be at least $(10^d+1)^2$ which has more than $2d$ digits, a contradiction. Thus we need $10^d+1$ to be the multiple of the square of some prime for $\overline{NN}$ to have the possibility of being a perfect square. $d=11$ is the smallest value for which this works.
This just might be useful.  If $n$ is even, $10^n$ is a perfect square, so $10^n + 1$ cannot be a perfect square.