On binomial coefficients modulo squares of primes

We give elementary proofs for the Apagodu-Zeilberger-Stanton-Amdeberhan-Tauraso congruences $$\sum _{n=0}^{p-1}{\displaystyle \left(\genfrac{}{}{0px}{}{2n}{n}\right)}\equiv {\eta }_p\mathrm{m}\mathrm{o}\mathrm{d}{p}^2,$$ $$\sum _{n=0}^{rp-1}{\displaystyle \left(\genfrac{}{}{0px}{}{2n}{n}\right)}\equiv {\eta }_p\sum _{n=0}^{r-1}{\displaystyle \left(\genfrac{}{}{0px}{}{2n}{n}\right)}\mathrm{m}\mathrm{o}\mathrm{d}{p}^2$$ and $$\sum _{n=0}^{rp-1}\sum _{m=0}^{sp-1}{{\displaystyle \left(\genfrac{}{}{0px}{}{n+m}{m}\right)}}^2\equiv {\eta }_p\sum _{m=0}^{r-1}\sum _{n=0}^{s-1}{{\displaystyle \left(\genfrac{}{}{0px}{}{n+m}{m}\right)}}^2\mathrm{m}\mathrm{o}\mathrm{d}{p}^2,$$ where $p$ is an odd prime, $r$ and $s$ are nonnegative integers, and $\eta_{p}= \begin{cases} 0, &$\text{if }$p\equiv0\mod 3;\ 1, & $\text{if }$p\equiv1\mod 3;\ -1, &$\text{if }$p\equiv2\mod 3 \end{cases}.$$