Regular expression for binary words with 1 in the middle
The intersection of your language with $0^*10^*$ is $\{ 0^n 1 0^n : n \geq 0 \}$, which can be shown to be non-regular by reduction to the more well-known $\{ a^n b^n : n \geq 0 \}$. This shows that your language indeed isn't regular.