 Solution

Every $x\in {\mathbb{Q}}_2^{\times }$ has a unique form $2^{n}u$ with $n\in \mathbb{Z}$ and $u\in {\mathbb{Z}}_2^{\times }$. Its square class first records $n\mathrm{mod}2$. An odd 2-adic unit is a square precisely when it is congruent to $1$ modulo $8$: necessity follows by squaring an odd integer, and sufficiency follows from Hensel lemma applied in its standard $2$-adic square-root form.

The odd residues $1,3,5,7$ modulo $8$ therefore give four unit square classes. Together with valuation parity this yieldsFor example, the classes of $-1$, $2$, and $5$ form a basis of the square-class group of the 2-adic numbers.