Negative range for binary cross entropy loss?
One thing you can do, is forcing your labels (${-1,1}$) to be ${0,1}$ using this simple linear transformation:

\begin{equation*}\hat{y} = (y + 1) / 2\end{equation*}

This way, -1 maps to 0, and 1 maps to 1.

For practical purposes, you can either change the outputs and labels of your model directly (before applying the original BCE), or slightly change to BCE loss function according to the linear transformation:

\begin{equation}H_p(q) = -\frac{1}{N} \sum_{i=1}^N \frac{(y_i+1)}{2} \log\left(p\left(y_i\right)\right) + (1-\frac{(y_i+1)}{2})\log\left(1-p\left(y_i\right)\right)\end{equation}

Notice that I changed only the coefficients, assuming that the probability function $p$ only returns meaningful values for ${-1,1}$

NOTE: obviously, all of this is relevant only for the binary cross entropy case.