Saharon Shelah (elaborating on Solovay) proved that it is equiconsistent with ZF that there exists a model of ZF+DC (DC=dependent choice, ) in which all sets of reals have the Baire property. And indeed all subsets of any Polish space have this property.
Now if $G$ is a second countable (=metrizable) profinite group, it is a Polish space (indeed a Cantor), and if $H$ is a non-open finite index subgroup, $H$ cannot have the Baire property, by a standard argument (it would have to be meager, which is absurd).
Hence it is consistent with ZF(+DC) that there is no discontinuous homomorphism from a second countable profinite group to a finite group.
On the other hand, I don't know if one can reduce the general case to the second countable one, e.g. by finding a countable intersection of open normal subgroups inside $H$.
