score:9
Since there seems to be a little disparity between the questions title and body, let me clarify that this answer addresses the last part of the question(emphasis mine):
To what religious group does the Flushing Remonstrance refer with the word "Egyptians"? If Islam, why is this listed distinctly from "Turks"? Or is it not associated with religion?
I believe that from the three groups that you find listed on this document, the reference to "Egyptians" is a reference to Africans in general, related to the use of biblical genealogies to quantify what earlier generations had observed about the dispersion of ethic groups. You can see these divisions in the images of maps on the wiki page discussing the Generations of Noah The group from Africa are called the sons of Ham. (An 1823 map from the same wiki page shows this concept applied at a world wide level.)
The quote from the OP's cited article cuts off just before the relevant part, "as they are considered sonnes of Adam". This has to do with biblical genealogies being used to explain ethic groups, but the Dutch group implying that all were equal as all were sons of Adam. It was a shot against the pro slavery groups. Some groups used the biblical Curse of Ham as a validation for the enslavement of Africans. This anti-slavery sentiment is echoed a little later in the text of the document (emphasis mine):
The law of love, peace and liberty in the states extending to Jews, Turks and Egyptians, as they are considered sonnes of Adam, which is the glory of the outward state of Holland, soe love, peace and liberty, extending to all in Christ Jesus, condemns hatred, war and bondage.
A direct statement of the opposition of slavery which is meant by including Egyptians (Africans, Sons of Ham) in this list. The phrase 'sonnes of Adam' is in direct repudiation of those in support of slavery in New Amsterdam.
So the Flushing Remonstrance not just about freedom of religion.
(I know this seems a little shy on sources, I was asked to convert some comments I made last night into an answer, so it was a little underdeveloped.)
For those of you who might wonder if this theory is corroborated anywhere, I found a 2004 book which seems to agree with my position:
Black and White Manhattan: The History of Racial Formation in Colonial New York City, by Thelma Wills Foote
This author on page 49 brings up the some notions, that the Flushing Remonstrance was not just about religion:
'That complaint raised fundamental objections against the enslavement of black Africans and their descendants in the Dutch overseas colonies.'
The author also brings up the Sons of Ham issue, a bit more clearly then my answer does, so take a look at the link.