Take $C: y^2 = x^6 + b$. There is a $(2,2)$-isogeny $J_C \to E_b^2$ corresponding to the factorization $\{x^2 - \beta_1, x^2 - \beta_2, x^2 - \beta_3\}$ of $x^6 + b$, where the $\beta_i$ are the roots of $x^3 + b$. While these $\beta_i$ may only be defined over some extension of $\mathbb{F}_q$, the set of factors is rational over $\mathbb{F}_q$, and therefore the isogeny is defined over $\mathbb{F}_q$.

I think a good reference for this (over number fields) is Section 3 of Howe, Leprevost, and Poonen's Large torsion subgroups of split Jacobians of curves of genus two or three.

Edit: this answer originally ignored an important twist.

Take $C: y^2 = x^6 + b$. There is a $(2,2)$-isogeny $J_C \to E_b\times E_{b^2}$ with kernel corresponding to the factorization $\{x^2 - \beta_1, x^2 - \beta_2, x^2 - \beta_3\}$ of $x^6 + b$, where the $\beta_i$ are the roots of $x^3 + b$. While these $\beta_i$ may only be defined over some extension of $\mathbb{F}_q$, the set of factors is rational over $\mathbb{F}_q$, and therefore the isogeny is defined over $\mathbb{F}_q$. The curve $E_{b^2}$ is a cubic twist of $E_b$, so if $b$ is a cube in $\mathbb{F}_q$ then you can compose to get an $\mathbb{F}_q$-isogeny to $E_b^2$.

I think a good reference for the isogeny construction (over number fields) is Section 3 of Howe, Leprevost, and Poonen's Large torsion subgroups of split Jacobians of curves of genus two or three.