It's never negative, and it's 0 only when y and ˆy are the same. Note that minimizing cross entropy is the same as minimizing the KL divergence from ˆy to y.

What does negative binary cross entropy loss mean?

When binary cross entropy predictions are negative, it is because the true values are not [0,1]. In my case I was using [-1,1]. The model does not fail, but produces negative value.

Can binary cross entropy loss be greater than 1?

Mathematically speaking, if your label is 1 and your predicted probability is low (like 0.1), the cross entropy can be greater than 1, like losses.