Generally speaking, Spanish cuisine is pretty good for dairy/egg free eating. Food is cooked in olive oil rather than butter. There is Paella of course, which contains no dairy or eggs. Normally on offer is meat or fish cooked a la plancha, which means on the grill.
As in most places in Europe, big cities such as Barcelona and Madrid have lots of vegan restaurants. You can find them using the Happy Cow app and Google Maps. Recently on a trip to Barcelona I found a vegan and gluten-free bakery close to my hotel - I was in heaven!
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Eggs are popular in Spain. A mixed salad normally comes with boiled egg. Tortilla and Revueltos are common egg-based dishes that you may need to avoid.
The vegetables in Greece are delicious! Commonly cooked in olive oil they will be no problem if you are vegan or avoiding dairy.
If you eat meat, then sadly moussaka is not an option for a dairy-free meal but there are lots of tasy alternatives such as stifado (beef stew), or souvlaki (chicken or pork kebabs). If you are staying on an island then there will be plenty of locally caught seafood.
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Some meals come with a side of Tzaziki which is a dip made of yoghurt. Greek salad generally contains feta cheese, however there is often an alternative such as a tomato and cucumber salad.