Most of us were there. I was out of the hand and will rely on others to fill in better details but the essence of the hand was that Player A is on a flush draw that gets there on the river. Board is paired. Player A bets 100. Player B raises to 300. Player A pushes. Player B makes the obligatory call. Player A shows King high flush. Player B shows Tens full of Fives.
I think at this point you really need to consider your opponent. The more experienced they are, the more they understand what a paired board REALLY means. To some n00bs, having three of a kind is such a gorgeous looking hand they will go to the mat with it, usually with no regard for their kicker. But when it comes to old salts, even having a house on a paired board is reason for concern. A while ago I made a crying call at Ryan's for the rest of my stack to call his raise of my river bet when the board pairs Eights on fifth street. His Eights full of Kings trumped my Eights full of Threes. I know Ryan well enough that I don't HAVE to call there. If he has trip Eights then he's smooth calling there if he calls at all. If he's raising then he has a boat and I had the nut low boat so what am I beating?
The problem is, even if you happen to read your opponent as a green player, there exists the possibility that they fell into a fluky full house or may have been sitting on a set the whole time. I've even run into a hand where I made trips on the river and a recreational player thought that she lost because she only had a set of lower rank which of course boated up when the board paired.
This is all of course passes over the fact that a King high flush isn't even the nut flush and that there may not even be any boats out there and you could still be behind Ax suited.
So anyway, Dr. Weak Tight says you check/call a value bet on the river.