Is there a way to upgrade my existing ships?

So to upgrade your fleet you first need to update the ship design, you can go into the ship designer in the top left and add the Coilguns. What I think you may have done wrong is created a new ship design and added the Coilguns there instead of updating your current Papillon-class Corvettes. When you change your ship design make sure you are modifying the Papillon-Class corvette and not creating a new class.

If you want to change the name of the class you can do that in the ship building screen, if you create a new class it won't change the existing class. Once you have update the Papillon-class and/or renamed it the upgrade fleet button should be Colored.

Right click in your home world with the ship selected and there will be a message to upgrade the fleet to the latest designs.