Greg is the pet version of Gregorovic. It is unlocked by inspecting the Faceless mask, which can be obtained as a drop from the boss. The odds of receiving the item are 1 in 2,000 (normal mode), or 1 in 1,000 (challenge mode), both with a threshold of 400 (explanation of drop mechanics). This is not affected by the increased drop chance reward from reputation.
This article on a pet has an associated dialogue page. Click here for a transcript of possible conversations with this pet.
Greg's first conversation is a reference to the movie The Mask.
Greg's third conversation is a reference to the graphic novel V for Vendetta.
Greg's fourth conversation is a reference to the TV show The Mighty Boosh.