“守口如瓶”常见的英文表达有 “keep one's mouth shut like a bottle” （较为直白、口语化的表达） ，更常用且更地道的表达是 “keep one's mouth closed/tight” 或 “be tight-lipped” 。
Keep your mouth closed about this matter. It's a secret.（对这件事要守口如瓶，这是个秘密。）
He's been tight-lipped about his future plans.（他对自己未来的计划一直守口如瓶 。）