Look, I've had my issues with Egwene, but try to look at it from her perspective.

First off, from books 7-12 Rand is going insane. And not only that, he's become uncaring and unfeeling. Inhuman. Both Nnneave's trip into Rand's mind after healing Naeff in ToM and his own viewpoint in Veins of Gold confirm this.

He almost kills Tam in a fit of rage. Even Tam himself says, "Unstable?" Tam asked. "Nynaeve, that boy is right near insane. What has happened to him? I understand what battle can do to a man, but ..."

Nyneave is with him for a lot of these episodes, and the balefire one a Natrim's Barrow. Egwene talks with her. As have other AS and eyes and ears to witness varying episodes of behavior from Rand that are, to put in mildly, troubling. Temperamental, distrustful, arrogant, and erratic are the words the eyes and ears have used to describe Rand. (ToM, The Amyrlin's Anger, Siuan POV). And it's hard to argue they are inaccurate.

Second, the idea that Egwene has never been a good friend to Rand is just silly.

Despite all this negative information she is getting on Rand, she still doesn't turn on him in ToM. Even though she seems set to before their meeting. Thinking of him as a criminal. And before all you Egwene bashers jump in, this is a view even Rand himself shares: "I should have wished for such a convenient set of backs on which to heap the blame for my crimes." ToM, The Amyrlin's Anger, p83, HB.

Egwene thinks to herself: "There didn't seem to be madness in his eyes. She knew those eyes. She knew Rand." "Light", she thought, "I'm wrong. I can't think of him only as the Dragon Reborn. I'm here for a reason. He's here for a reason. To me, he must be Rand. Because Rand can be trusted, while the Dragon Reborn must be feared."

Still, breaking the seals is a bridge too far for her, despite her thoughts that Rand could be trusted with the fate of the world (ToM, HB, p. 84). But she wants to change his mind, not oppose him violently.

And most of her opposition comes from the dream she has misinterpreted (him hacking the globe, which, I believe, deals the with future of the Aiel, the 23 Stars, being, I think, a WO+chief for the 11 clans remaining, + 1 for the Shaido and + 1 for the Jenn). Regardless, I don't think it has to deal with the Seals.

She doesn't turn on him in ToM, and she hasn't the whole series. Let me count the ways. In EotW she overcomes her initial reaction to Rand being able to channel and hugs him anyway, telling him she doesn't care. In tGH she helps Rand escape from Fal Dara. Throughout the whole series, she never once tells anyone what he is until it is already common knowledge. She stands up/lies to Elaida on this score numerous times in the Tower, and even to Moiraine as well. She and Elayne try to help Rand learn to channel in tGH, and to detect a woman's channeling. She conspires with Elayne to pass him off to her best friend. Even when leading the rebels, she still can't believe Rand would forcibly bond sisters. And these are just the ones off the top of my head.

And this idea that Rand doesn't like or respect Egwene? It's foolishness. No doubt, they've butted heads on some issues (bonding AS, for instance). And been wary of each other from time to time. But what does Rand thinks when Siuan tells him that Egwene is an incredible Amyrlin? "I should have expected nothing less."

Rand isn't just counting on Egwene to oppose him. Which is only natural. Break the seals? That would free the DO. Everyone one knows if the DO breaks free, everything is doomed. How often has a character said, "If the DO was free already, we'd all be dead!" Thus, the notion of breaking the seals sounds crazy to anyone who hasn't been inside Rand's head for the entire series. Or almost everyone (Nynaeve, Perrin, Elayne, etc). He's counting on her to be able to unite the rest of the world and get them there in less than a month. And she does deliver.