(1) With $H$ the Heaviside function, define $Log(x+i0)=\ln(\vert x\vert)+i\pi H(-x)$ and $$T_1=\frac{1}{x+i0}=\frac{d}{dx}(Log(x+i0))=pv\frac 1{x}-i\pi \delta_0(x).$$ It is easy to see that $WF T_1=[0]\times (0,+\infty),$ so that $WF T_1+WF T_1$ does not meet 0. Then there is no difficulty to define $T^2$ say as $$\langle T^2,\phi\rangle=\lim_{\epsilon\rightarrow 0_+}\int\frac{\phi(x) dx}{(x+i\epsilon)^2}.$$
(2) Let us consider a smooth hypersurface $\Sigma$ of $\mathbf R^d$ defined by the equation $f(x)=0$ with a smooth $f$ such that $df\not=0$ at $f=0$ and let $\delta_\Sigma$ be the Euclidean measure on $\Sigma$. Then $$T_2=pv\frac{1}{f}-i\delta_\Sigma$$ can be squared. The reason is the same than for the previous example, since $WF T_2$ is the positive conormal of $\Sigma$. A point $(x,\xi)\in WF T_2$ iff $$x\in \Sigma\quad \xi =\lambda df(x) \text{ with \lambda >0}.$$ Then of course, if $(x,\xi_j)$, $j=1,2$ are both in $WF T_2$ then $$\xi_1+\xi_2\not=0.$$