Min had a number of viewings about Rand, some of which seemed connected. The two I mentioned above were not specified as being connected, but we've already seen that, at least once, they were definitely connected.

LoC, Chapter 41: "...And there was something else, something new, an aura of yellow and brown and purple that made her stomach clench. "Aes Sedai are going to hurt you. Women who can channel, anyway. It was all confused. I'm not sure about the Aes Sedai part. But it might happen more than once. I think that's why it seemed all scrambled."
LoC, Chapter 46 (in regards to Perrin and Rand): "When you two were together, I saw those fireflies and the darkness stronger than ever...But with the two of you in the same room, the fireflies were holding their own instead of being eaten faster than they can swarm, the way they do when you're alone. And there's something else I saw when you two were together. Twice he's going to have to be there, or you... If he's not, something bad will happen to you. Very bad. It will happen if he is not there, but nothing I saw said it won't because he is. It will be very bad, Rand."

The first time Aes Sedai hurt him happened when he was kidnapped and brutally tortured (and Min was kidnapped as well). I've always thought that, if Perrin had been with him, it might not have happened at all, but the general consensus is that he was, indeed, there for Rand by rescuing him and prevented the Aes Sedai from hurting him even worse. Regardless of the view you take on the matter, the fact is that the two viewings were related - Perrin had to be there for Rand when Aes Sedai hurt him.

Personally, I've always believed the second time Perrin should've been there for him was when he met Semirhage. Had Perrin been there, Min would've been forced to stay behind - Rand would've wanted Perrin to be there for a double whammy of ta'veren - and Rand would not have been more focused on protecting Min than on self preservation. That has always been my opinion, but now I think I might have been wrong...

This meeting that Egghead arranged (due to Rand manipulating her) has the potential for disaster if she refuses to see reason (and when does Egwene ever see reason if it means admitting anyone else is right and she has been wrong?). All indicators point to her refusing to see reason and demanding that Rand obey her (she has gone batshit crazy about everyone obeying her without question and has turned into a mirror image of Darth Rand from before his epiphany). She even goes as far to think that he'd better do what she wants so he doesn't "force" her hand. I think she fully intends to attack him if he makes it clear that she isn't his boss. Enter Perrin...

Perrin supports Rand. He - this guy who is known for contemplating minor things longer than it takes most people to make major life decisions - knew as soon as he heard what Rand planned that Rand is correct. That alone speaks volumes, as does the fact that Cadsuane, who spent all of her time up until ToM calling Rand an idiot and treating him like a drooling mental deficient brat, never once tried to deter him once he told her his plan.

Then there is Egwene who, if Rand declared that grass was green, would swear that grass was, instead, blue, and that he clearly needed her ruling him. There is no way she'll back down on this. She would oppose him until the end of time because it hurts her ego to let anyone think, for even a moment, that he is her equal or better. She has to be seen as superior - as does the WT - so there is no way she'll stop opposing him after gathering everyone she could by telling them that they must oppose him. So how is this connected to the viewing? Easy...

When it becomes clear to Eggy that Rand won't submit himself to her authority (for South Park fans, every time I think of Egwene I think of Cartman with "RESPECT MY AUTHORITIE!"...it's so fitting...), Egwene will get it in her head that she has no other choice than to force him to her will. Otherwise, she may look like she's submitting to him. So, she will shield him and try to take him into authority to force him to obey her (possibly by publicly beating him to show that she has control of the situation). This is the second time to which Min was referring - Aes Sedai, particularly Egwene, may hurt him a second time. But her viewing about Perrin is also applicable - Perrin isn't going to allow it. I think there is a reason that he so quickly accepted Rand's decision as rational and correct, opposing Egwene's snap judgment that Rand is wrong and must be stopped at all costs - he is the one with the power to stop Egwene .

Yes, Cadsuane might oppose her and try to point out that Rand is right, but Egwene would never listen to her. She also ignores anything Nyn says, instead choosing to believe that Nyn was brainwashed. Perrin is a different story, however. Yes, he's a man (and all men are useless morons to be controlled by women, in Egwene's POV), but she used to respect him somewhat (as much as she is capable of respecting anyone, anyway) and he recently made her look the fool in TAR, where she thought she was reigning queen. Throw in the ta'veren effect from him (I think Rand would intentionally quell the effect with her to give her enough rope to hang herself and the WT - if she refuses to listen to him after others do and tries to do the typical AS thing and force him to her will, the people gathered there will see she's just like the rest of the AS and that the WT can't be trusted at ALL) and she may be forced to pull her head out of her ass for once. It may be the slap she needs to change her attitude about Rand and get it out of her head that she has to control him. Or not.

Either way, I think that the second time Rand may be hurt by Aes Sedai is at this meeting and believe that it is Perrin's presence that will stop it before much can happen.