No. A counterexample is (essentially) given by $$ u(z) = -\log |z+i\delta|, \quad\quad z=x+iy, y\ge 0 . $$ This function is harmonic and positive near $z=0$, and $|u(x+iy)|\le |u(x)|\in L^2(-1/2,1/2)$, so $$ \int_{-1/2}^{1/2} dx\int_0^{2\eta} dy\, u^2 \lesssim \eta , $$ as required. However, $|u(i\eta)|$ is not bounded as $\delta,\eta\to 0+$.
By rotating and (slightly) rescaling this, we obtain a counterexample in your setting.