The answer is Rule 12 and 14.
If on the river all three players check then Player A and B may muck without showing, if so, then player C is last man standing and rule 14 does not allow player A and B to ask to see the winning hand.
rule 12 is just about showdown order but if A and B decide to muck player C still has the right to ask to see the hand but they would be live and he could be beat.
And as Nick says, C does not have to show in the first scenario. (theoretically, another player at the table that was dealt into the hand could ask to see player C's hand, but that would be rude) Also, there are some places that are using a house rule (TDA does NOT have this rule) that says you HAVE to show your cards to win the pot. I think that one is silly but a lot of places are doing it even though it is not a TDA rule.