I'm not 100% sure if this was a cash game or a tournament, since you called it both "live game" and "tourney" but I'll assume it was a cash game.
I'll also assume that even if there isn't a physical "button", the passing of the deck is treated as if there was one, since that is generally how it happens.
In games which use a dealer button, it generally depends on the house rules whether the button is "live" (where the button always moves to the next player) or "dead" (where payment of the BB advances around the table, and the SB and button are placed based on that, whether a seat is occupied or not).
No matter which of these variation is being used in this game, unless the player who busted out is not intending to rebuy (or it's a freezeout, and they can't) it should be correct for you to post the SB on the hand as described.
If the player was leaving, and a "live button" is being used, I would expect the deal to move to you, and the blinds would be posted in front of it.