Which parts of the fish are squeezed to get oil for fish oil capsules, and what is the fish used for after it's been pressed?

The whole fish. The whole fish is used through a complicated process that separates the water/fat from the solids. The fat is then separated from the water. The remainder of the fish is used as "fish meal" in a variety of ways from fertilizer to animal feed. Go to http://www.Fao.Org/docrep/003/x6899e/x6899e04.Htm for a full explanation.
It's complicated. In short, the entire fish is used in the making of fish oil and fish meal. During the processing, the solids and liquids are separated, and then the oils are separated from the liquids. The oils are then purified. Check out this website from the food and agriculture organization of the united nations for a more detailed explanation: http://www.Fao.Org/docrep/003/x6899e/x6899e00.Htm.
Usually the meat. The meat or muscle tissue is used to obtain the fish oil. I am not sure what is done with the byproducts of manufacturing, but would contact the manufacturer.