I'd answer this except there's such a big difference between having the nut flush and the nut straight (even if it's the broadway).

Nut flushes can't be split and they only have to worry about a paired board river (and the rare str8 flush - but, as I said, that's rare). There is no such thing as a weak nut flush on the turn. These hands can be slowplayed occasionally on the turn against the right opponents, especially if you have loose callers left to act behind you.

Naked braodway straights however are simply the most dangerous nut hands you can possibly hold. They split more than any other straight (because more people play KJTx than 865x). If the river is another broadway card, you'll either almost always split or get beat by a full house. Especially if there are two of a suit on a board, this is the most common way to get freerolled. If I have good re-draws or am short stacked I'll usually push. If I'm deep, naked, and OOP I'm nervous as all hell.