If the Jinchuuriki and their tailed beast work together they can basically break every genjutsu put upon them since one of the counters against genjutus is to have the victim's chakra flow disrupted. This makes them mostly immune to them.
A victim's chakra flow, if disrupted enough, can break the caster's influence. This "Genjutsu Dissipation" (幻術解, Genjutsu Kai, Literally meaning: Illusion Technique Dissipation) or "Genjutsu Reversal" (幻術返し, Genjutsu Gaeshi, Literally meaning: Illusion Technique Reversal)[7] can be done by the victim themselves, (assuming they realise they're under a genjutsu's influence) or an ally who emits chakra into them;[8] jinchūriki who have formed a good relationship with their tailed beast prefer the latter method whenever they need it.
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