> both successor states
Why would there be only two?
Sure. You could abolish the Union entirely, and just have states. I don't think it's obvious that that would be a terrible thing - either for USAians or for the rest of us. My sense is that a lot of USAians don't really want a Union. I suspect that many USAians think the Union was effectively a coup.
The question remains: how do you spread the nukes around?
My guess is that the residents of Wyoming could do without nukes. They have a lot of land-based nukes, that serve no defensive purpose (for Wyoming); they just make Wyoming a target. I wonder which state might decide that they need nukes, to defend themselves from someone outside North America?
Second question: how you divide up the military equipment that isn't nuclear - ships, tanks, rockets, planes etc. All that equipment carries a heavy cost - it has to be maintained, replaced, tested etc.
The Lord's our shepherd, goes the psalm When Alabama gets the bomb.
~ Tom Lehrer, "Who's next"
Well, the USSR broke up without too much fuss about who got the bombs; as far as I can see, only Russia was interested. But I can imagine a Divided States of America in which both successor states have nukes. I suspect they'd be reluctant to attack one-another. They might even both choose to disarm, depending on the nature of the break-up.