In FTL, the "Save & Quit" options is disabled during combat. If I am in the middle of a battle and have to quit, will I lose all my progress, or can I still continue from some recent point, like right before the battle started?
If you quit the game your data will not be saved, you'll need to start over from the beginning. Source: tested it in game.
Your save and quit is a one shot deal. It works by writing your current game data to a file called continue.sav in your FTL directory.
This file only exists if you do "Save & Quit" from the menu. As soon as you load up your game and continue your quest, your data is loaded and the file is immediately deleted. So any interruption in the application, like quitting in a battle, while playing before another "quit & save" will result in a permanently lost quest.
You could always back up continue.sav, but this is considered cheating by many and not the intent of the developers. But it's your game, do as you wish.
The latest version of FTL adds the ability to save at any time, so the new answer is that yes, your progress is always saved, even if you quit during a fight or some other danger.