After reviewing this episode, there's a line that clarifies why Miller wants the bombs diverted, and explains why Holden agrees to divert the missiles.
After watching the next episode, it becomes clear why Johnson went along with this plan.
Here's the conversation between Miller and Holden where Miller suggest diverting the missiles (added emphasis):
Miller: You gotta stop following Eros. It's gonna defend itself, and my money ain't on you.
Holden: I can't do that. We'll lose it for good.
Miller: Forget the goddamn missiles. She'll dodge those just like she did the Nauvoo, or something a lot worse.
Alex: Hoss, I'm losing visual...
Miller: Your crew is gonna die for nothing. Back off now. Find a way to divert those missiles.
Holden: And then what?
Miller: What? I'll try to reach Julie, if there's any part of her still left. Get her to stop this rock.
Miller isn't concerned that the missiles will hit them, or what might happen to them eventually if they keep going, he's concerned that Eros will take control of the missiles. They could be redirected at the Rocinante, Earth, or anywhere else. Given that Eros has already done things that they would have otherwise thought impossible, Holden takes this possibility seriously.
Holden then sends Johnson a short message saying that it's vital that the missiles be redirected, but not giving any reason.
Johnson initially refuses, pointing out that Earth will think that they're stealing the missiles. He is then told that the Rocinante has already lost Eros, and that there's no chance of the missiles hitting their target.
After taking this in, Johnson orders the missiles diverted.
From the events in the next episode, Paradigm Shift, we can surmise why he made that decision. He figures that if the missiles don't hit their target and are diverted, before reaching Eros' approximate position or after, then Earth will think he's broken faith and stolen the missiles either way, therefore
he may as well just actually steal them. Nuclear missiles can be useful. He then proceeds to steal 30 missiles by screening them from Earth so that the abort codes can't be used. The rest of the missiles are remotely disabled.