Quote 1
"Will you promise to always protect her?" [Eve] asked. She wasn't asking me. She was asking Denny, and I was merely Denny's surrogate. Still, I felt the obligation. I understood that, as a dog, I could never be as interactive with humanity as I truly desired. Yet, I realized at that moment, I could be something else. I could comfort Eve when Denny was away. I could protect Eve's baby. (6.20-1)
This moment between Eve and Enzo is about two things. First, Eve might be suggesting that she won't always be around to protect her daughter, so she's asking Enzo, and by proxy, Denny, to do it for her. Second, Enzo steps up to the plate here, realizing that while he can't take control of being human, he can take control of his own life, in whatever small way he can, and do what Eve asks of him.
Quote 2
"Get me through tonight," she said. "That's all I need. Protect me. Don't let it happen tonight. Enzo, please. You're the only one who can help." (23.109)
This is another instance when Eve relinquishes control of her own life to someone else, once again Enzo. She asks him to watch over her because she's too afraid to fall asleep and succumb to the terror that's lurking for her in the night. Fortunately, Enzo makes an excellent guard dog, just not against demon zebras.
Quote 3
"Today is the first day I am not dead," Eve said, "And we're having a party." (27.12)
It's important that Eve says that she "is not dead," not that "she doesn't feel dead." It could mean that she has felt dead up until this point, and now that she knows she will die, she isn't dead, because death isn't the end. That seems ironic—and really sad.