Now leaving aside for a moment the fact that Cook did not come here as a mendicant migrant hell-bent on fleeing religious persecution back home - but rather a representative of the world's then-mightiest Imperial power on a globe-crossing voyage of discovery - it's an interesting notion.
Albeit a point which is better phrased in another other image in circulation which depicts a Native American telling a Pilgrim in no uncertain terms "sorry, but we're not accepting refugees".
The problem for those on the liberal left (or even evangelical right) who're lining up to make the argument in favour of accepting Syrian refugees by using these memes, is that history did not exactly work out favourably for those peoples generous enough to host (whether willingly or otherwise) foreign migrants. As another pixellated Indian tells us: "We took in refugees ... look what happened to us!"
Similar problems abound for images which suggest US Senator Ted Cruz wouldn't be in America (and thus a leading light of extremist right-wing politics there as well as an alleged Presidential contender) were it not for America's previous acceptance of refugees and political exiles from Cuba. It's only a matter of time before a similar meme based around John Key pops up noting he wouldn't be Prime Minister were it not for New Zealand accepting Jewish refugees fleeing the Holocaust and its aftermath in the middle years of the last century.
In each case, these are not exactly strong arguments in favour of accepting thousands of Syrian refugees into our midst.
But regardless of how one feels about the present-day Prime Minister, it would seem something of a stretch upon our conventional morality - knowing what we now know about the conditions they were fleeing - for us to throw up our walls, bar the gates, and block Jewish refugees seven decades ago from safe haven on our shores ... even if that's *exactly* how American (and other) populations seemingly overwhelmingly felt about Europeans fleeing Nazi persecution in, say, 1938.
So why, then, have I seen so many New Zealanders stating skepticism about allowing Syrian refugee settlement here (or, for that matter, elsewhere in the Western world) in recent days?
The answer's simple: Fear.
Following on from what happened in Paris over the weekend, news outlets lost no time in pointing out that one of the terrorists responsible for Friday night's atrocity may have managed to slip into Europe undetected in the guise of a Syrian refugee - a horrendous irony if true given the fact that fleeing these men and their actions is exactly what's driving the mass-migration of the Syrian people in the first place.
People are, therefore, perhaps understandably spooked about the idea that allowing refugees to enter into our own countries - quite apart from the usual issues that are part and parcel with playing host to new arrivals - may potentially mean welcoming a terrorist threat into our own homes. That acting on decent humanitarian impulses by allowing refugees in carries with it the salient risk of them bringing with them exactly the same evils bedeviling their homeland.
Yet - as a recent Economist piece points out - of the nearly three quarters of a million refugees whom America has accepted since 9/11 ... none have been arrested on any domestic terrorism charge.
More interestingly, there is now speculation which suggests that the Syrian passport found at the site of the Stade de France attack - and which was used to push the narrative of ISIS infiltration via refugees - may not in fact have come from a Syrian refugee at all.
French police appear to believe the passport itself was a fake, manufactured in Turkey. This doesn't, in and of itself, disprove allegations that the man referred to in media reports (rightly or wrongly) as Ahmed Almuhamed snuck in to Europe hidden amidst genuine refugees (although you have to wonder if the notoriously dangerous Mediterranean crossing and arriving in a sinking ship is really the best way to infiltrate personnel).
But it is interesting - even suspicious - that "Almuhamed"'s passport was found at the scene of the attack. One might be tempted to ask what sort of terrorist or covert operative carries clearly identifying personal information on them unless they want it to be found; and French officials have also raised issues around the passport's placement at the scene - suggesting that the passport's presence itself was an intentional act of communication from attackers to target nation.
If true, then the strategy being deployed by ISIS is both obvious and insidious. By making it seem like every Syrian refugee or Muslim migrant is a potential terrorist ... they don't just make potential targets feel continuously, constantly afraid. They capitalize upon that fear when Western hatemongers - the tabloid presses and the far-right aspiring politicos - then broadcast this message to their own readers, audiences and constituents.
How does stirring up hate against Muslims help ISIS? Simple. It feeds into the "us-and-them" dynamic that's integral to the group's world-view - and helps to alienate people living in target countries.
France has long had a problem on this front, with French Prime Minister Manuel Valls directly stating that the divide between France's communities was one of "territorial, social and ethnic apartheid".
These are strong words - the severity of which helps to encapsulate the depth of feeling that's evidently helped to contribute to France producing a number of French-born extremists in recent times that have carried out attacks in Paris ranging from the Charlie Hebdo episode through to the present circumstances.
Starting to see why ISIS wants you to demonize Syrian refugees and Muslim migrants yet?
It appears to be a prime factor driving a very small number of young men into some seriously twisted arms, beliefs and actions. And the best thing about home-grown recruits is you don't have to expend effort infiltrating them across borders - they already live in the country, in the city being targeted.
But that isn't a Pauline Hanson-esque set of reasons to eschew engagement with the demographics being talked about. Instead, quite the contrary.
As Australian commentator Waleed Ali pointed out in his excellent analysis of the subject, to do so would be to "help these bastards" by playing directly into ISIS's hands.
Admittedly, memes suggesting that the present waves of Syrian migration are in any way comparable in severity if not outright danger to successive - and invariably bloody - efforts at European colonialism of other continents are arguably only a very small part of that, if part of that at all.
But at times like these, when emotions are running high and knee-jerk reactions (often delivered swiftly and sharply to somebody else's solar plexus) seem to be all the rage ... it's worth taking pause to stop and critically consider what we're saying and to whom.
Ali's right. This is the time to stand together - not to fall apart. The extremists on all sides - whether white supremacist or sand-strewn lunatic - will hate that.
Which is arguably a pretty great start on it being *exactly* the right thing to do.