Step 1 :
Go to a Pet administrator
Step 2 :
Talk to the NPC and click on the highlighted option
Step 3 :
It will open this tab
Step 4 :
Click on the Upgrade Tab
This will show you the requirements to craft either a Hercule or a Phoenix. As you can see you need a couple of different pet eggs. These pets are considered rare pets and have a hefty respawn timer when after they are tamed or killed, some of them won't respawn for a full 48 hours.
These are the pets you need to tame with the locations they tend to walk / fly around.
when you've tamed all of them you go back to the pet administrator to craft your herc or nix
NOTE: Keep in mind that these monsters move around and thus won't necessarily be on the exact spots the maps indicate. They usually move in between those waypoints from one point to the other.