Many people are concerned that squatting low - that is, with your knee at an acute angle, rather than just to the point that your thighs are parallel to the floor - can cause knee injury. I've seen a few sites (including the one I linked yesterday) that claim otherwise, but the very fact of seeing so many denials made me wonder whether anyone is making a credible case that deep bends are risky.
So I found this article, which argues pretty extensively (with multiple references) that squats are safe for most people, even people with various types of knee injuries, and in general have a protective effect on knees. My squats have been causing me a little bit of knee discomfort and "fuzziness" (both of which go away by the next day), and the article suggests I should do them shallower if that's the case, until I build up strength. (I've been nervous about the knee sensations, but not sure whether they were injuries or just soreness like you often have in your muscles after lifting weights.)
So I'll be keeping my squats shallower for now, but I hope to continue working up to higher reps and more depth over time as my knees adapt. The article also covers spinal concerns, but I won't be worrying about my spine until I actually start squatting with a barbell (or at least a bar).