What affection is required for companion quests

UPDATE:

Apparently at some point, a patch has been applied to the game that now allows companions affection to be negative. See other answers for details. (I haven't confirmed this or I'd put the details in this answer.)

ORIGINAL:

No. Your companion's Affection can not go below the initial starting value, indicating neutral.

If you're on an RP server, feel free to role play their hatred towards you by staging a fight with them yourself, then never summoning them again after you betray them and let them die in a fight you tell them to start as you run away laughing at their demise.

Otherwise, (seriously) the benefit of having high Affection with a particular companion increases their chances of crafting and mission successes, yields better quality results and opens up better gear.

As a rough rule of thumb, every 750 affection is worth roughly a 1% reduction in completion time on crew skill tasks.

Edit: Comments indicate that the first 450 points of companion affection are worth 2% off the top, and that further affection follows the 1%/750 formula. I haven't seen the exact data set, but this does fit the patterns I've seen closely enough to be plausible.