Assuming that by "inaccessible Jónsson" you meant a regular limit cardinal of uncountable cofinality which is Jónsson; then using the arguments of [1] (Theorem 15 p.115), we have

> if $\mathbb{P}$ is c.c.c. and $\kappa$ is Jónsson then for any $V$-generic $G\subset \mathbb{P}$, $V[G]\vDash$ "$\kappa$ is Jónsson".

In particular, if $\kappa$ is Jónsson, and $G\subset \mathbb{P}=\mathsf{Fn}(\kappa^{+}, 2)$ is $V$-generic, then 

> $V[G] \vDash $ "$\kappa < 2^{\aleph_0}$ and $\kappa$ is Jónsson." 

hence $V[G] \vDash \neg \diamondsuit_\kappa$ and $\kappa$ is Jónsson. Moreover, If we started with $\kappa$ which was a regular limit cardinal then the same holds for $\kappa$ in $V[G]$.

[1] <cite authors="Devlin, Keith J.">_Devlin, Keith J._, [**Some weak versions of large cardinal axioms**](http://dx.doi.org/10.1016/0003-4843(73)90010-7), Ann. Math. Logic 5, 291-325 (1973). [ZBL0279.02051](https://zbmath.org/?q=an:0279.02051).</cite>