Yes, both are fine. I don't know what the other two commenters mean by 'spoil' meaning that you did something wrong. It's not. A spoilt appetite is simply an appetite that is not as strong as it should be.

Yes, you can use both. As Georgie said, "spoil" implies that you did something wrong, like eating junk food right before dinner. I would also add that "lose" could imply you're not hungry because you feel sick.

Yes you can use both. 'Spoil' implies more that they're doing something wrong