Good question

It is clear, that we can consider only relative refraction index n0/(n0 +/- delta n). Simplest way is to induce birefringence in glass sample by mechanical force. You can obtain n + delta n for compression and n - delta n for stretching.

People have constructed complex layered metamaterials that exhibit a negative index of refraction. However, other than those exotic examples, optical materials do not have negative indices of refraction. near resonances (like a resonant absorbing state) the index of refraction can fall below 1, but I don‘t think that is relevant. So, in regards to birefringent materials you should think in terms of two normal indices of refraction each > 1. What is important for birefringence is the difference in the indices. When you write the Jones matrix you could leave both true indices of refraction. However you are welcome to factor out a common term. You could factor out the fast term leaving a one and what looks like a positive index or you could factor out the slow leaving a one and what looks like a negative index or you could factor out the average and leave what looks like one positive index and one negative. However, remember reality includes the factored term. As Aleksandr pointed out, if the birefringence is created by perturbing an isotropic medium either by stress or electric field or whatever people generally like to factor out the original unstressed index of refraction so that the birefringence is in term of the applied field. In that case very often the indices go opposite directions and often not symmetrically. In any case, it usually doesn’t matter how you write it (if you are doing something time dependent and keeping track of phase, you’ll need the factored term anyway) and the birefringence effects will depend only on the difference in the indices.