A great many of these individuals are given over to visual memes and short, punchy, "take it back" messages. But then, this is the nature of Twitter - visual memes and short, punchy, messages. It is not designed for "substantial" argument in any measure, and seems more appropriate as a proving ground for mutations on logical fallacies.
It is perhaps a measure of the times that, aside from the usual political cartoons and memes, the political figure I see oft quoted by conservatives is Thomas Paine. Two quotes from Paine seem the most used (abused?), namely:
It is also a measure of the times in which we live that the first quote is likely spurious, having been lifted from the writings of an American "anarchist, environmentalist and novelist" named Edward Abbey. As for the second, I suspect it is also spurious. I ran a search here and here, and found nothing. Well, nothing, that is, older than 1997 or so, when I suspect this quote likely originated....at least on the internet, if Google can be believed (a stretch, I know). Jefferson is equally likely to abuse. One meme which just appeared on my Twitter feed states "The greatest danger to American freedom is a government which ignores the Constitution." While attributed to Jefferson, it is equally a spurious quote, which one site believes appeared in 2015.
All this is not to say that genuine quotes from any given person do not appear on Twitter - which it IS to say is that the individuals posting these memes seem not to care whether they are genuine or not. It is sufficient that the meme expresses some sentiment of the moment, and bolsters that sentiment with a reference to authority.
Recently, on the site "Minding the Campus," Patrick Deneen published a post entitled "How a Generation Lost Its Common Culture." Therein, Deneen describes experiences with his students and their lack of knowledge of Western History (broadly defined). Deneen laments that, while the students are pleasant enough and may have accidental knowledge of a variety of topics:
We have fallen into the bad and unquestioned habit of thinking that our educational system is broken, but it is working on all cylinders. What our educational system aims to produce is cultural amnesia, a wholesale lack of curiosity, history-less free agents, and educational goals composed of content-free processes and unexamined buzz-words like “critical thinking,” “diversity,” “ways of knowing,” “social justice,” and “cultural competence.”Most of the essay resonated strongly with me, in teaching myself and in knowing many other professors. However, what particularly made me think was the "wholesale lack of curiosity." While Deneen describes students where he has taught (Notre Dame, Princeton, Georgetown), the lack of curiosity he describes is widespread. I think that this lack is sourced in our basic education system (K - 12), and definitely continued by post-secondary education.
Once one considers the widespread lack of curiosity, I think one is led inexorably to the profusion of spurious quotes and logical fallacies which pervade Twitter (as well as the explanation for the increasing popularity of Twitter, Vine, Snapchat, and other "instant" forms of communication). If modern education is completely set against older history, it is also engaged in eradication of more recent history - time spans measured in months and years, not necessarily decades. How else to explain the rise of the modern politician, who can, without effort, explain (away) beliefs, statements, and actions completely contrary to those espoused in the moment? The answer is that, with some exception, few care about what was said a month, a year, a decade, ago. As Deneen notes, "They have learned exactly what we have asked of them – to be like mayflies, alive by happenstance in a fleeting present."
This occurrence affects both conservatives and progressives, though progressives seem to embrace it, and conservatives are unsure as to how to stop it. I think that the oft-spurious citations to authority are an attempt by conservatives to hearken back to some cultural heritage - to some idea of limitation and history. Unfortunately, too often, being educated in the school system designed to "to sand off remnants of any cultural or historical specificity and identity that might still stick to our students" (Deneen), many conservatives are left without enough curiosity and attention to even check the sources, to read The Federalist Papers, the Constitution, famous court cases, etc. And therefore, we are left solely with expressions of sentiment in their raw form, without expansion of those sentiments formed by centuries of deliberation and debate.
So, in the end, we see (once again) C.S. Lewis's predictive powers in action. To quote from "The Abolition of Man":
The final stage will have come when “humanity” has obtained full control over itself. “Human nature will be the last part of Nature to surrender to Man.” The ruling minority will have become a caste of Conditioners, people “who really can cut out posterity in what shape they please.” From this moment onward, the human conscience will work the way humans want it to work – that is, the way wanted by the Conditioners. What are the Conditioners going to want in conditioning our consciences, and, indeed, how are they going to want anything? Human ideas about good and evil, duty and taboo, are among the things for them to decide about and therefore cannot serve as a ground for their decision. All motives for human action have become objects of choice and manipulation by Conditioners; so the Conditioners themselves are left without any motives. Unless they stop moving and acting at all, they must become prey to any force that just happens to put them in motion – in other words, to irrational, natural impulses. And since their power is perfectly effective, the human race will for the rest of its existence be subjected to such forces of nature as happen to have acted upon the Conditioners. Man’s conquest of Nature will have brought about Nature’s conquest of Man: the Abolition of Man.