The phrase "near squeak" refers to a narrowly avoided disaster or failure. There are various synonyms that can be used to express the same idea, such as close call, close shave, narrow escape, hair's breadth, or narrow margin. Additionally, one can use other expressions like a nail-biting moment, heart-stopping moment, or edge-of-your-seat moment to convey the same sense of a moment of intense anticipation and eventual relief. Regardless of the specific word choice, all of these synonyms capture the feeling of narrowly avoiding danger or failure and the accompanying intense emotions that come with it.