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She just won't share her opinion.
Literal
She [topic-は] her own opinion [object-を] [just doesn't-なかなか] try-to-say.
なかなか + negative is a productive pattern meaning 'not easily, just won't, hardly ever.' なかなか alone can mean 'quite, considerably' in positive contexts, but with a negative predicate it flips to 'won't readily.' ~ようとしない is the negative of ~ようとする — 'won't try to, refuses to.' Stronger than just being slow to share an opinion: she actively resists.