If aliens invaded, I'd still call the United States the United States. I'd also still call Boston Boston, North America North America and Earth Earth. I don't really see any reason to cooperate with invaders by eradicating place names.
While this could change if you survived the invasion and then the country was rebuilt 20/30 years later. They were marking the 2 year anniversary of the invasion. It's not like people are going to forget *THAT* fast and for the most part around the world people who grow up in countries are nationalistic and take pride in their country.
The survivors have been through horrible events so giving them some bit of normalcy in the United States, a President, and an election system is a bit of relief from their world being turned upside down.