Can I add extra eggs instead of xanthan gum?

If you want a flaky crust: No, you can't. A flaky crust is made from small pieces of gluten sheets formed when part of the dough reacts with the water, separated by melted fat blobs. The idea of making a crust with coconut flour instead of normal flour is already risky: it will give you some kind of crust, but nowhere as good as a normal flour crust. Maybe it is worth doing it in the case of food allergies, but not just for the flavor.

Provided that you want to make a flaky crust, you need the non-fat dough part to stick together well in sheets, and the fat dough part to form an uniform soft paste. If you use coconut flour, you need a protein to make the non-fat dough part stick together. As the wheat flour is missing, you can't use gluten. The only substitute I know of which indeed does make something sheet-like is xanthan (also, it does bind, but not as well as proteins do). Other proteins, like the ones in egg whites, will indeed bind your coconut flour, but the amount needed will be so high you will have dominant egg whites creating tough, leathery "layers", not crispy at all, and tasting very much of eggwhite. And I say "layers", but they won't be layers the way you would get them with flour, it is more likely to disperse with the fat part in irregular shapes, not flakes. This is if you only use egg whites; yolk will make the matter worse here.

The fat part of the dough will be even worse. The coconut flour won't absorb the fat, building a smooth paste. It will mix with the fat, but will stay in its way when it melts, preventing it from creating the very thin pockets needed for flaky crust.

Conclusion: I doubt that you can make any kind of flaky crust without flour. Xanthan may give you a not-so-good approximation; just eggs will give you nothing.

If you want a shortbread crust, you are better off. You don't need the gluten structure in the water part and the smooth starch of the fat part. There are lots of shortbread recipes which include substantial amounts of nut flours for flavor, although they tend to stop at 1:1 nut flour to wheat flour at most. I guess that you can go with nut flour all the way if you must, but then expect a crust even more crumbly than usual. Xanthan is a good ingredient to hold it together in this case, although it won't give the texture of a wheat crust. Additional eggs will work too (slightly beaten, but not whipped), but they will make it less of a crust, they will wet it a lot and push it in the direction of batter. Try using whites only, and drop the water altogether. For better, more tender texture, add starch, for example corn starch. And because you are already moving into batter direction, you are risking a dense, soggy crust. Add just a little big of baking powder to prevent it (not the same amount as for cake, maybe 1/4 of it), this will make it a bit more bisquit-like.

you can certain convert the recipe uscing coconut flour, but remember that the coconut flour is "wetter" the the other, so you can use coconut flour but perhaps you should add less butter, or less water... and why vinegar? i have never known that we can use vinegar in dessert recipes.. sorry for bad english, but i'm italian!