Perhaps it goes without saying that in both series Game 2 was a must-win for the teams that lost the Game 1. Or nearly without saying, as a few teams have lost the first two games in a best-of-seven series and still come back to win. But I think in these particular cases the White Sox and Astros simply had to win their second games.
In the case of the White Sox, if they'd lost the first two games, they'd have been faced with the prospect of traveling to Orange County for three games, and needing to win twice to stay alive when the most likely outcome would have been two wins for the Angels and the ALCS never even returning to Chicago for a Game 6.
In the case of the Astros, being down by two games wouldn't have been all that far from being down by three games. Why? Because they're going to be serious underdogs in Game 4. Here are the pitching matchups for the rest of the NLCS, along with each starter's place in the National League's ERA rankings: