Morally speaking, I would answer in the affirmative; yes they should still be able to claim compensation from the exploitative colonial powers of their recent past. Legally, probably not, as independence is what it sounds like; going it alone as an independent national entity means being entirely responsible for past, present and future realities. Countries like Australia, which perpetuated colonial activity in New Guinea, tend to maintain high levels of international aid toward their former colonial territories; and this can be viewed as some sort of continuing reparation.
Imposing western values on indigenous communities out of a sense of racial and cultural superiority is now seen to be morally wrong. The fact that these actions were also only masking the real activities, like mining or agricultural exploitation for profit, is doubly reprehensible. Colonial activities by European powers through Africa and Asia, in the eighteenth and nineteenth centuries in particular, were accompanied by military force to enforce the property rights of the colonial interests over the indigenous peoples. Almost, slave labour conditions often disenfranchised these local people from a share in the wealth being generated in their country using their natural resources. These people deserve compensation but it is highly unlikely they ever will receive it from the companies and families who exploited them.
Colonial western powers brought diseases and degenerative behaviours to many previously idyllic natural environments, which were populated by indigenous peoples. The progress of these super powers changed the world for these denizens of natural paradises. Their lives were now seen as lesser than those of the economically dominant white skinned people; they were savages, little more than animals. Christianity played its part in this cultural besmirching of ‘so called’ native peoples by informing them of their damned and godless state. Missionaries self righteously contributed to the eradication of indigenous religious beliefs and cultures; ignorantly casting many of these people into a ‘no man’s land’ between western and indigenous worlds.
The writings of Joseph Conrad began an anti-colonial awareness in literature, which became a flood of post-colonial writers from all over the globe. Writers such as VS Naipaul, Salman Rushdie, Edward Said, Leopold Senghor, Chinua Achebe, Leon Damas, Bharati Mukherjee; to name just a few, have all contributed greatly to a post-colonial awareness of the injustices committed upon the indigenous peoples of so many countries by colonial bully boys. Generations of injustice are not wiped out easily and independence is, unfortunately, a two edged sword. Colonialism, also, has not completely disappeared as seen by the behaviour of China in Africa recently.