Six years later it's hard to know if this is still a question of interest. For the record (since I stumbled across the question), the answer is yes and it's a consequence of a more general phenomenon.

The main theorem of a paper of Stevens and Warren (Thm 5 in https://arxiv.org/abs/2102.05446) states that for any convex (or concave) functions $f, g:\mathbb{R}\rightarrow \mathbb{R}$ we have $$ |A+B|^{38}|f(A)+g(B)|^{38} \gtrsim |A|^{49}|B|^{49} $$ where $\gtrsim$ hides factors of $\log(|A|)$.

Then an answer to the original question follows by setting $A = B$ and $f = g$ with $f(x) = x^2$. This gives, with $S = {a^2 : a \in A}$ $$ \max\{|A+A|,|S+S|\} \gtrsim |A|^{49/38} $$ i.e. an exponent of 1.28947...

Six years later it's hard to know if this is still a question of interest. For the record (since I stumbled across the question), the answer is yes and it's a consequence of a more general phenomenon.

The main theorem of the paper of Stevens and Warren (Thm 5 in https://arxiv.org/abs/2102.05446) states that for any convex (or concave) functions $f, g:\mathbb{R}\rightarrow \mathbb{R}$ we have $$ |A+B|^{38}|f(A)+g(B)|^{38} \gtrsim |A|^{49}|B|^{49} $$ where $\gtrsim$ hides factors of $\log(|A|)$.

Then the answer to the original question follows by setting $A = B$ and $f = g$ with $f(x) = x^2$. This gives, with $S = \{a^2 : a \in A\}$ $$ \max\{|A+A|,|S+S|\} \gtrsim |A|^{49/38} $$ i.e. an exponent of $1.28947\ldots$