Hera's fears are not specifically addressed as such in Greek mythology. However, her actions suggest that she feared the loss of her husband and power.
Ironically, Hera marries Zeus only after he her rapes her as an effort to hide the act, and they infamously do not get along. Hera even plays major rolls in organizing several revolutions against Zeus, though none of them are successful.
Zeus has several affairs. Despite their marital problems, Hera is always quick to take revenge on the women in question. Hera also plays an active role in ensuring that some of Zeus's sons do not usurp his throne by driving them insane.
Hera has great power over animals and likes to use them as a mechanism for revenge. She often turns her enemies into animals or sends animals to kill them. Although she is the goddess of marriage and fertility, she is not known for her matronly skills. When she had a child without Zeus in response to his having a child without her, she banned the child from Olympus because she feared its ugliness. Hera's jealousy leads to a feud with Heracles, one of Zeus's sons born to another woman as the result of his affairs. Hera even tries, unsuccessfully, to stop his birth. Some myths eventually depict Heracles and Hera eventually making peace.