The context of this claim: how did the New England colonists view the land they were settling?
The Eastern seaboard of America had four major strains of colonists: Quakers and Cavaliers, who emphasized living in harmony with the native Americans, Scots-Irish in the South, who did not care much about natives at all, and the famous Puritans/Separatists, whose entire worldview was very strongly religious. Of these groups, the comparison of Indians to Canaanites/Amalekites which is being claimed here would be coming solely from the Puritans.
The Puritans were indeed making a metaphor between their journey from England to New England, and the journey of the Israelites out of Egypt and into Palestine. However, they were entirely aware that this was only a metaphor. There have been some Protestants who literally believed that the Old Testament's message applied to them (notably the Boers of South Africa), but the Puritans were very much aware that they were living in the present day.
We can find proof of this in the Cavalier Thomas Morton’s book The New English Canaan (1637). Morton was imprisoned and banished by the Puritans, and referred to himself as a Canaanite. He described the natural bounty of New England as far closer to the appearance of temperate Canaan than the desert of Israel, and proposed that natives and colonists ought to reject the self-appointed "chosen tribe" (Puritans) and their "New Israel" (Plymouth) and live together in a multicultural New Canaan, which is the origin of the present-day city of New Canaan, Connecticut.
Morton was not renouncing Christianity or Englishness by taking up the Canaanite cause. Nowhere in his book do we see a hint that "Israel" might want to go to war with him. Rather, he was using the Puritans' metaphor against them, in a way that he was probably expecting to hit home and get on their nerves.
With that background, we can now answer the question.
Cotton Mather is one of the most famous Puritan preachers. He lived during King Philip's War, which was a legitimate all out war between Englishmen and Native allies against other Indians. During the war he gave sermons encouraging Puritan forces to take up arms, though it might be against their first instinct, for the sake of the community.
From "a discourse delivered unto some part of the forces engaged in the just war of New-England against the northern & eastern Indians":
Let me mind you, that while you Fight, Wee'l pray. Every good man
will do it, in secret and in private every day; and publick
Supplications also will be always going for you. We will keep in the
Mount with our Hands lifted up, while you are in the Field with your
Lives in your Hands, against the Amalek that is now annoying this
Israel in the Wilderness. It was the Watch Word which a Battel once
Commenc'd withal Now for the Fruit of Prayer!
From the context we can understand that this was a handy combination of the prominent Israel metaphor with the actual situation of war. It was not all Indians which were compared to "Amalek" but only the enemies of the Plymouth settlement. In peacetime, as far as I can tell through searching, the Indians were not ordinarily considered as akin to Canaanites or Amelekites or otherwise fair game for victimizing. The Puritans sometimes made allies, sometimes made enemies, with the natives. Practical political situations, not sense of religious mission, determined the type of rhetoric that was used.
Conclusion: No, the early New England settlers never actually assumed the "right" to treat Indians this way. They enjoyed comparing themselves to Israel in sermons, which included analogizing enemy Indians to "Amalek", but that did not enter into their legal or political calculations.