= Solution
Every $x\in\mathbb Q_2^\times$ has a unique form $2^nu$ with $n\in\mathbb Z$ and $u\in\mathbb Z_2^\times$. Its square class first records $n\bmod2$. 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 yields
$$
\mathbb Q_2^\times/(\mathbb Q_2^\times)^2
\cong(\mathbb Z/2\mathbb Z)^3.
$$
For example, the classes of $-1$, $2$, and $5$ form a basis of the <square-class group of the 2-adic numbers>.
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