DNA study links indigenous Brazilians to Polynesians
Indigenous people that lived in southeastern Brazil in the late 1800s shared some genetic sequences with Polynesians, an analysis of their remains shows. The finding offers some support for the possibility that Pacific islanders traded with South America thousands of years ago, but researchers say that the distinctive DNA sequences, or haplogroups, may have entered the genomes of the native Brazilians through the slave trade during the nineteenth century.
Most scientists agree that humans arrived in the Americas between 15,000 and 20,000 years ago, probably via the Bering land bridge linking northeastern Asia with what is now Alaska. But the precise timing and the number of ‘migration waves’ is unclear, owing largely to variations in early Americans’ physical features, says Sérgio Pena, a molecular geneticist at the Federal University of Minas Gerais in Belo Horizonte, Brazil.
One broad group of these Palaeoamericans — the Botocudo people, who lived in inland regions of southeastern Brazil — stands out, having skull shapes that were intermediate between those of other Palaeoamericans and a presumed ancestral population in eastern Asia.
http://www.nature.com/news/dna-study-links-indigenous-brazilians-to-polynesians-1.12710
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Polynesians didn't reach Rapa Nui (Easter Island) until about 1200 AD. As that is the easternmost point of known Polynesian culture, it is probable that any contact between them and South America would have happened after that. Vastly too late to have been the initial colonists of the continent.
I wouldn't rule out some limited trade between the two... both of goods and genes. But as the Rapa Nui islanders seem to have rather quickly exhausted their boat-making resources, any such network would have been very brief.
As for the land bridge, there are two strong cases for its existence and viability as a pathway; One, geology. Glaciation is pretty easy to suss out, and there is in fact a corridor that seems to have been glacier-free through Alaska and western Canada. Second is biological - there is a very strong continuity of species between North America and northern Eurasia, extinct and extant. Unless you want to tell me large numbers of genetically-diverse bison swam the whole way? 
some humans no doubt migrated by sea (in fact, this is how the Aleut and Inuit peoples got where they are; their migration from Eurasia was within the last 4,000 years) but there's nothing wrong with a Berengia foot crossing.
This might help explain some things, such as why the Dene peoples have a few remaining relatives in Oregon and why the earliest confirmed remnants of Native American presence on our continent have been found along certain areas of the Northwest coast so far, and not Alaska.
The Bering land bridge theory may indeed help explain the origins the Inuit and related peoples, such as the Tlingit, and perhaps some of those communities in British Columbia, and maybe even adjacent areas of Washington, as well, but not of the Athabascans and their Navajo cousins, and perhaps many others as well.
I, for one, would like to see more in-depth research into this whole matter. 
