When can you use の to replace a noun?

Statistically speaking, I think it's true that 学生じゃないの usually refers to a state ("not being a student"), whereas 白いの usually refers to an object ("white thing"). However, the correct meaning largely depends on the context.

Please remember that の meaning one usually replaces a noun representing an inanimate object (i.e., 物). Sometimes の can also represent a human being, but it's not very common, and it sounds rough and/or condescending (similarly to 奴). For example, it's possible to say "学生じゃないのが学校に来るな!" or "もっと強いのを呼べ!". But it's safe to avoid this usage if you are a beginner.

の also works as a nominalizer. Both 学生じゃないの and 白いの can represent a state. For example, you can say 学生じゃないのを悔やむ ("to regret not being a student") and 髪が白いのを心配する ("to worry about someone's hair being white").