Some of them are BoP, some of them are BoU. Thankfully, ALL of them can be caged. So it's not absolutely necessary that we'll need to have each profession to collect these pets (time to make friends with your local crafter! ;P)
Alchemy (all BoP but cageable - 700 skill, requires player level 106)
Enchanting (all BoU and cageable - any skill level, requires player level 106)
Engineering (BoU and cageable - 700 skill, requires player level 108)
Fishing (BoU and cageable, requires reputation to purchase - any skill level)
Archaeology (BoP but cageable - requires player level 110)
It should be noted that crafting in Legion is not quite the same as in previous expansions. Many recipes are not simply learned from a trainer or from a dropped plan.
Instead, there are quite a few recipes that can only be obtained by completing profession quests. All of the alchemy pets, one of the enchanting pets, the engineering pet, and archaeology pet all require the completion of a quest or a series of quests.
Here's a summary for the profession pets that will require questing (please correct me if I get any of this information wrong!)
Alchemy - Alchemists will need to complete a quest chain to unlock the purchasable recipe for Wild Transmutation. Creating this has a chance to teach a number of other transmutations. The one pet collectors are aiming for is Transmute: Meat to Pet. This creates an item that, after 7 days, has a chance to drop one of the 3 new alchemy pets. It should be noted that Legion transmutation recipes have a one day cooldown.
Enchanting - Excluding the Enchanted Cauldron and Enchanted Torch, both of which are learned from purchased recipes, the Enchanted Pen recipe requires completing a quest chain, starting with the quest An Enchanting Home (requires player level 106).
Engineering - I haven't been able to find much information on how to learn the recipe to craft this pet, except that it may come from a quest or event called Trigger Happy.
Archaeology - The archaeology trainer will have quests for level 110 players every 2 weeks. Completing these quests will teach the player 'rare solves', one of which is the new pet.