I couldn't get all the way through this story on the military island 1,000 miles south of the Indian coast, but if you're interested in studying and understanding American and British imperialism and/or South Asia, I would strongly suggest taking an glance. Wikipedia link provided for your use as well. Enjoy:
In the early 1970s, Britain quietly handed over Diego Garcia, one of its furthest-flung territories, to the US military. In the process, a remote island idyll was simply emptied of its people, allowing for the creation of a place so secret that no journalist has been allowed to visit—a place that became a key staging post in George W. Bush's wars in Afghanistan and Iraq, and a crucial node in the CIA's rendition system, a "black site" through which at least two high-value suspected terrorists were spirited, far from the prying eyes of international law.
You need a baseline theory of property rights before you conclude that anything's been done wrong to the Chagossians--so what that it's an Island--if they came in as hired labor, renting accomodation, I don't see why they have a right to stay once no longer welcome.
I am not an expert on Diego Garcia or its residents; however, I would note that your phrase "once they are no longer welcome" undercuts your point about the importance of "a baseline theory of property rights." It poses the question - who decides when someone or something is welcome, and who decides when it is not - i.e. questions of social power. In that sense, it is not hard to side with the Chagossians and I would go far as to say that most of the people who write on and read this blog are probably attuned to questions of imperialism and social power. You could carry this further to question the relatively ahistorical notions of property rights that are presented by neoclassical economics and explore the historical development of them - critical legal theory does this, as far as I know, as does some heterodox economics (e.g. primitive accumulation).
However, the reason I posted the articfe was more to get a sense of the history of the island and the miltiary presence there, so it's interesting that you focused on defending the expulsion of the people who lived there rather than anything else in the article.