How the positive and negative frequencies are defined depends on your choice of vacuum, and in quantum field theory the vacuum is unambiguously defined. You can sort of think of these as representing particles and anti-particles. When you quantise a field you get positive frequency and negative frequency parts.

Now onto the second bit, and I approach this with trepidation. NB 'positive' and 'negative' doesn't mean 'particle' and 'anti-particle' (for what it does mean see below), and the black hole will radiate equal numbers of particles and anti-particles. This entangled system can interact with the black hole and split, and the interaction guarantees that the emerging particle will be the positive one. Instead the pair form an entangled system where it's impossible to distinguish between them. When a pair of virtual particles are produced there isn't a negative energy particle and a positive energy particle. I'll attempt both, but I'm outside my comfort zone so if others can expand or correct what follows please jump in! The first is to explain what Brian Greene means, and the second is to point out that the 'particles being swallowed' explanation is a metaphor and isn't actually how the calculation is done. There are two ways to approach your question.

Black particles radiation island