So, as many as you know, the hatch system has been implemented recently. I'm going to show you how hatching your pet egg will work, and some tips.
The first thing you want to do is get your Rowell Egg into your bag, if it is locked, unlock it. The pet egg will be under the Collectibles section. Head to Log Out>To Island>Your Island>Edit Mode. Then go to Isle Object. Tap on the button that looks like a Colon, select the egg you want to use, and click the button "Use".
Did you get this error message? This means that you already have 3 pets, and if you want to use the egg, you will have to get rid of one of your existing pets. I got rid of my Colon, never used him much anyway o-o
If you've changed your mind, you can always push "no".
Congrats you've got a little tiny baby Rowell. Unlike non-egg pets, it'll grow up over time and become very big ^_^ You won't have to feed him multiple times to make him your pet, like other pets. Once you open the egg, he's all yours.
Rowell's favorite food is going to be liquid (satisfied or very satisfied). Meat is also acceptable (eating silently). I'll look more into the other types of foods, on the JP wiki it doesn't have much information, but I recommend that you feed your pet liquids for now.