While Zeena has a plethora of negative traits, it is the "asthmatic" silence in which she engages what would definitely qualify as the hardest thing for Ethan to tolerate.
In chapter IV, we find evidence which suggests that silence is something that bothers Ethan quite a lot. In fact, the only time Ethan actually shows frustration in the novel is found in this chapter, when his mother becomes ill and falls into a constant silence.
...when his mother fell ill the loneliness of the house grew more oppressive than that of the fields. His mother had been a talker in her day, but after her "trouble" the sound of her voice was seldom heard, though she had not lost the power of speech.
Ethan would actually react wildly to his mother’s silence
Sometimes, in the long winter evenings, when in desperation her son asked her why she didn't "say something," she would lift a finger and answer: "Because I'm listening."
This will occur again with Zeena, nee Zenobia Pierce, Ethan's cousin and future wife.
When she first comes to nurse Ethan’s mom, Zeena is described as someone who is seemingly jolly. She releases Ethan from the burdens of the nursing process, and her voice is what Ethan claims kept him from going crazy in the house.
His gratitude to her, and his fear of living alone in a silent, cold farm are the factors that prompt Ethan to propose to Zeena and marry her. However, things begin to fall apart and, a year after the marriage, Zeena, too, “went silent.”
She only spoke to complain, and Ethan seldom answered her, so she would keep to herself. However, at other times, she uses her silence to push Ethan’s buttons since she knows what effect silence has on him.
When Zeena remains quiet, she causes Ethan to feel fear and guilt regarding his feelings for Mattie. Additionally, being silent while witnessing the dynamics between Mattie and Ethan helps her disguise her true emotions, which also makes Ethan wonder what she may or may not suspect about them. Essentially, Zeena's tendency to remain mute makes Ethan anxious and nervous throughout the novel, although he also wonders if Zeena's silence could possibly be a sign of insanity.
Of late […] her silence had begun to trouble him. He recalled his mother's growing taciturnity, and wondered if Zeena were also turning "queer." …[H]e himself knew of certain lonely farm-houses in the neighborhood where stricken creatures pined, and of others where sudden tragedy had come of their presence… [H]e felt the chill of such forebodings.
Therefore, out of all the many annoying traits that affect Zeena’s personality, the silence that she uses to her own benefit really bothers Ethan. He cannot discern what she is thinking. He cannot figure out her non-verbal cues. He simply does not know what to make of it. Much like it was with his mother, this silence is designed to drive him nearly insane.