Is square numbers written in binary a regular language?
Here is a high-tech solution. Denote your language by $L$. Szalay showed that
$$
L \cap 10^*10^*1 = \{ 10^n10^{n+2}1 : n \geq 0 \} \cup \{110001, 1000010001\}.
$$ 
From here it's pretty easy to show that $L$ is not regular, by reduction to $\{a^nb^n : n \geq 0\}$.