A conjecture about algebraic values of $(-q;\,-q)_\infty/(q;\,q)_\infty$

Yes, it is always algebraic, because it is a modular function evaluated at a CM (complex multiplication) point.

"$(q;q)_\infty$" is $q^{-1/24} \eta(\tau)$ where $q = e^{2\pi i \tau}$, so "$(q;q)_\infty / (-q;-q)_\infty$" is a root of unity times $\eta(\tau) \, / \, \eta(\tau+1/2)$, which is modular for some congruence subgroup of ${\rm SL}_2({\bf Z})$, i.e. a rational function on some modular curve $X$. If $q = e^{-\pi \sqrt x}$ then $\tau = (i/2)\sqrt x$ is an imaginary quadratic irrationality, and thus a CM point on $X$. It is known that every CM point is algebraic, whence the value of $\eta(\tau) \, / \, \eta(\tau+1/2)$ at the point is also algebraic.

The CM theory also provides further information about the degree of such algebraic numbers, their Galois group (always solvable), and conjugates (values of the same function at other CM points).