Properties of squares in $\mathbb Q_p$

Let $\mathbb Q_p$ be the field of $p$-adic numbers.

I know that for $p \neq 2$ an element $x=p^n u \in \mathbb Q_p^\times$ (with $n \in \mathbb Z$ and $u \in \mathbb Z_p^\times$) is a square if and only if $n$ is even and the image $\overline{u}$ of $u$ in $\mathbb F_p^\times$ (by the homomorphism $\mathbb Z_p \to \mathbb Z_p / p \mathbb Z_p \cong \mathbb F_p$) is a square.

And for the prime number $p=2$ I know that an element $x=p^n u \in \mathbb Q_2^\times$ is a square if and only if $n$ is even and $u \equiv 1 \pmod 8$.

Now, how to prove that for $p \neq 2$ the quotient group $\mathbb Q_p^\times / \mathbb Q_p^{\times^2}$ is isomorphic to the group $\mathbb Z / 2\mathbb Z \times \mathbb Z / 2\mathbb Z$ and that $\left \{ 1,p,u,up \right \}$ is a system of representatives, where $u \in \mathbb Z_p^\times$ is such that $\left ( \frac{u}{p} \right )=-1 $?

And that $\mathbb Q_2^\times / \mathbb Q_2^{\times^2}$ is isomorphic to the group $\mathbb Z / 2\mathbb Z \times \mathbb Z / 2\mathbb Z \times \mathbb Z / 2\mathbb Z$ and that $\left \{ \pm 1,\pm 5,\pm 2,\pm 10 \right \}$ is a system of representatives?

There is an isomorphism of groups $\mathbb{Q}_p^* = \langle p \rangle \cdot \mathbb{Z}_p^* \cong \mathbb{Z} \times \mathbb{Z}_p^*$. Therefore it suffice to determine $\mathbb{Z}_p^* / {\mathbb{Z}_p^*}^2$. If $p \neq 2$, look at the canonical homomorphism $\mathbb{Z}_p^* \to \mathbb{F}_p^*$. It is surjective, and by what you have said it pulls back ${\mathbb{F}_p^*}^2$ to ${\mathbb{Z}_p^*}^2$. In other words, it induces an isomorphism $\mathbb{Z}_p^* / {\mathbb{Z}_p^*}^2 \cong \mathbb{F}_p^* / {\mathbb{F}_p^*}^2$. Since $\mathbb{F}_p^*$ is cyclic, the latter has order $2$.