Spain
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.
Watch out for...
A mixed salad normally comes with boiled egg. Tortilla and Revueltos are egg-based dishes so you will want to avoid these.
Greece
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 fish.
Watch out for...
Some meals come with a side of Tzaziki which is a dip made of yoghurt. Greek salad contains feta cheese, however there is often an alternative such as a tomato and cucumber salad.