A recurrence with a double fixed point

Key identity. $u_{n+1}-u_n=u_n^2-u_n+\frac14=\left(u_n-\frac12\right)^2\ge0$. The sequence is always non-decreasing.

Case $|u_0|\le\frac12$. If $|u_n|\le\frac12$, then $u_{n+1}=u_n^2+\frac14\in\left[\frac14,\frac12\right]$, so by induction $u_n\le\frac12$ for all $n\ge1$. Non-decreasing and bounded above: it converges to a limit $\ell$ with $\ell=\ell^2+\frac14$, i.e. $\left(\ell-\frac12\right)^2=0$, so $\ell=\frac12$.

Case $|u_0|>\frac12$. Then $u_1=u_0^2+\frac14>\frac12$, and the sequence is non-decreasing from $u_1>\frac12$. If it converged, the limit $\ell\ge u_1>\frac12$ would be a fixed point, but the only fixed point is $\frac12$. So $u_n\to+\infty$.

Summary. $u_n\to\frac12$ if $|u_0|\le\frac12$, and $u_n\to+\infty$ otherwise.