Of course, in general it is not true that inequality between entropies implies the same inequality between $\ell^2$ norms. Although this is true for $n=2$, it fails already for $n=3$ (as the entropy $H(p_1,p_2,p_3)$ is obviously not constant on the level curve determined by conditions $\sum p_i=1$ and $\sum p_i^2=Const$).

Nonetheless, there is a deep link between these two quantities. In order to explain it, it is better to somewhat change the viewpoint. Namely, given two probability distributions $P=(p_1,\dots,p_n)$ and $Q=(q_1,\dots,q_n)$ (for simplicity I assume that all $p_i,q_i$ are strictly positive), the corresponding **Kullback-Leibler deviation** is defined as
$$
I(Q|P) = \sum p_i \log \frac{p_i}{q_i} \;,
$$
and the so-called **information energy** as
$$
\chi^2(Q,P) = \sum \biggl(\frac{p_i}{q_i}-1\biggr)^2 q_i \;.
$$
Both these quantities measure "closedness" of $P$ and $Q$ (although they are not distances) and are monotone invariants of the pair $(Q,P)$ (they do not increase under quotient maps). If $Q=U_n$ is the uniform distribution, then these quantities are, up to linear rescaling, precisely the entropy and the $\ell^2$-norm of $P$, respectively, as
$$
I(U_n|P) = \log n - H(P)
$$
and
$$
\chi^2(U_n,P) = n \sum p_i^2 - 1 \;.
$$

Now, the link between $I$ and $\chi^2$ is provided by the fact that in naturally defined infinitesimal limits the Kullback-Leibler deviation $I$ and the information energy $\chi^2$ coincide
and produce the **Fisher information metric**. One can read more about this in the corresponding wiki articles, in the old book *Statistical decision rules and optimal inference* by Čencov (AMS, 1982), or in more recent publications on **information geometry**.