What can I use to replace eggs in the recipe?
It looks like you're basically making a cheese custard for the center of the bread. Flax seeds aren't going to give you a firm, egg-like texture. Instead, I'd use something that already has that cooked-egg texture, like tofu, which also won't impart any flavor (so you'll need to tackle that issue separately).

Experiment with different preparations, but I'd start off with a silken tofu, then purée the tofu in a blender with a bit of cornstarch (to help it firm back up when cooked). Once the tofu has been processed you can mix in the shredded cheese. and throw the filling in.  You'll probably  have to do several attempts before you get the ideal preparations and keep in mind that it won't taste like the original thing, but may bear enough resemblance. 

EDIT: In response to the comment, I would say you could try using tapioca in a high concentration, dissolved in a bit of milk or cream. After cooking (hopefully) the tapioca will help to create the texture you want, and it will actually not affect the flavor. 

The other (obvious) option is to just leave out the eggs and not replace them with anything. It won't be the same, but neither will any of these substitutions, and it will likely still be good.