K.C.
We bake egg and dairy free all the time. Replacing eggs is simple. My favorite substitute is:
1.5 TBSP of oil
1.5 TBSP of water
1 tsp of baking powder
You'll need to double that since you are replacing two eggs. Every cookie, cake, muffin, etc that we have baked using that substitute has been great.
Do not cook with real eggs, as it will likely cause a reaction in the child with allergies.
As a mom whose son has allergies and is in school for the first time, I can say how much I appreciate that you are being cautious and looking for a way to include the whole class. : )