
Argentina is famous for it's meat. The beef is thought to be the best in the world, and the preferred way to cook the meat is asado (on the barbecue). In general the steaks come plain, so you don't need to worry too much about butter. A common condiment is chimichurri which is made of olive oil, parsley and garlic. Great for dairy-free! (but be careful if you avoid garlic).
If you are vegan, there are options available, especially in the bigger cities such as Buenos Aires and Cordoba. I found vegan restaurants, as well as health food shops selling tofu and other vegan produce. The supermarkets, such as Carrefour, have dairy-free milk and vegan burgers.
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Buenos Aires has a lot of pizza and pasta restaurants as there is a strong Italian influence. We looked at a couple of the menus but there were not many dairy/egg fee options.

If you eat seafood, a famous dish in Peru is cerviche, which happens to be dairy and egg-free. is raw fish prepared with lime juice and the fish is 'cooked' by the acid.
There is a Chinese influence in Peru and Chinese food is popular. One common fusion dish is Lomo Saltado: steak stir-fried with onion, tomatoes, spices, soy sauce and chips. It's served with rice.
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Pisco is the national drink of Peru, which is a type of brandy. However the popular cocktail Pisco Sour contains egg whites.

A popular dish in Costa Rica is Gallo Pinto which is rice with beans. This can be served for all meals of the day (including breakfast) and comes with various sides, my favourite was fried plantain.
There are lots of tourists visiting Costa Rica for the beautiful scenery and amazing wildlife encounters. You can therefore get a wide range of food in the restaurants, especially in the popular tourist sites, although a word of warning - eating out is quite expensive here! There is plenty of fish, and pork dishes seemed to be common. Not forgetting that since it's a tropical country, the fresh fruit is delicious!
Watch out for...
We found that Gallo Pinto is sometimes cooked with butter, especially when served in hotels for breakfast.