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She pressed her lips firmly together and resolved never to cry.
Literal
She [topic-は] firmly lips [object-を] closed-and never not-cry [quotative-と] resolved.
Three packed grammar points. (1) 唇を閉じる ('close one's lips') uses 閉じる, a more delicate verb than 閉める ('close [a door]') — appropriate for parts of the body and books. (2) 決して + negative is the universal-negation pattern: 決して泣かない ('never cry'). (3) 泣くまい uses ~まい, the negative volitional ('won't do, refuse to do, intend not to'), more emphatic and literary than the simple 泣かない. The whole sentence reads as the steeling of resolve in a moment of emotional control — a classic literary trope of dignified suffering.