Is $1\neq a\in Z(2.E_7(q))\cong Z_2$ a square element in $2.E_7(q)$?

Here is a general remark about whether a central involution $z$ in a finite group $G$ is a square : It is well known, and easy to derive from the orthogonality relations for group characters and properties of the Frobenius-Schur indicator $\nu$ that $z$ is a square in $G$ if and only if $\sum_{ \chi \in {\rm Irr}(G)} \nu(\chi) \chi(z) > 0.$ Since $\nu$ vanishes on irreducible characters which are not real-valued, the sum may be restricted to the real-valued complex irreducible characters of $G$. Note that the set $S$ of real-valued irreducible characters which make a positive contribution to the sum contains those $\chi$ which have $z$ in their kernel and $\nu(\chi) = 1,$ (contribution $\chi(1)$) and those $\chi$ which do not contain $z$ in their kernel and $\nu(\chi) = -1$ (contribution also $\chi(1)$). Any real-valued irreducible character $\chi$ of $G$ which lies outside $S$ makes a contribution $- \chi(1)$ to the sum. Hence $z$ is a square in $G$ if and only if $\sum_{ \chi \in S} \chi(1) > \sum_{ \chi \in {\rm Irr}_{\mathbb{R}}(G) \backslash S } \chi(1)$, where $Irr_{\mathbb{R}}(G)$ denoted the set of real-valued complex irreducible characters of $G$, and $S$ denotes the set of real-valued irreducible characters $\chi$ of $G$ with $\nu(\chi) \chi(z) = \chi(1)$.

However, I am not sure whether enough information about the character table and Frobenius-Schur indicators is available for the groups you are considering.

The answer is always, yes. Note that there are three classes of involutions in the simply connected version of the algebraic group $E_7$: the central involution $a$, an involution $t$ with centralizer of type $A_1D_6$, and the product $at$. If $a$ were not a square, then in the simple group $E_7(q)$, we would only see involutions with centralizer type $A_1D_6$. However, in the adjoint group we find centralizers of type $E_6T_1$ and $A_7$. This website: http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/23elts.html lists the classes of involutions in the simply connected and adjoint groups, or consult the 3rd volume of Gorenstein-Lyons-Solomon.

Notice that this yields explicit elements of the simple group that power to the centre in the central extension.