First, I think that referencing Nazi Germany in our current age is a failed effort, regardless of whether or not you're a 2004 Democrat or a 2012 Republican. It just doesn't work in our current society because even if it's an honest, sincere, and forthright comparison of a very specific instance (which, frankly, is rare imho) it still comes across as something else. References to Nazi Germany in any fashion almost always come off, and are likely intended, to be nothing but a guilt by association tactic of leveraging the emotional implications that go along with Nazi Germany within the American psyche.
You could be comparing and contrasting the transportation laws of Nazi Germany to that of the United States and it would still come off that you're attempting to discredit the United States laws by tying them to Germany's, because if they have ROADS like the Nazi's just think what else like the nazi's they could have! Oh, and you know what else the nazi's did? Gas Chambers! You don't support gas chambers do you!!!!!
The reality is that for the VAST majority of analogies where one uses Nazi Germany there is probably another country, regime, group, etc that you could use as the comparison instead that doesn't have the attached emotional baggage.
Now, as to his point about being told to "Shut up". In those speecific words, not so much....but I absolutely do think there's been a strong movement by both sides in this century to try and shout down and criticize those who speak up or take legal action against what the government or it's bodies are trying to do.
Obviously Ben Carson isn't the only presidential candidate that has felt this style of feeling sometime within the 21st century. Let me quote another Presidential Candidate speaking of it:
"'I'm sick and tired of people who say that if you debate and disagree with this administration, somehow you're not patriotic."
That was Hillary Clinton. If you'll recall the national conversation, especially from democrats at the time, the general notion was that accusations of lack of patriotism were meant to "silence" critics ... aka "shutting them up". In many ways, the Left's complaint that "any disagreement with Bush is called unpatriotic" mirrors the Right's complaint that "any disagreement with Obama is called racist".
Though indeed, the questioning of ones patriotism, or really their stance as an "american", as a means of "silencing" them is hardly just a right wing thing. Before you had Sarah Palin going on about "real americans" you had Barack Obama suggesting supporting a debt ceiling increase was "unamerican".
Many on the left also suggested the accusations of "supporting terrorists" or the like were clearly attempts to silence people from disagreeing with the Bush Administrations "war on terror" policies, and this continued for some time. Remember the indignation over John Aschcroft suggesting disagreement with the BA anti-terrorism policies were supporting terrorists and the claims that it was trying to silence their critics. And again, what is "silencing" but a nice say of saying trying to "shut up" their critics? You have that kind of thing continuing on today as well. You had members of this current administration comparing those who wanted spending cuts to go with a debt cieling raise as "terrorists" and people with "bombs strapped to their chests". You had individuals who disagreed with Democratic legislative attempts and sought to stop them being compared to arsonists and anarchists.
Now personally, my THEORY is this attempt to compare and label ones politics opponents in extremely negative ways as a method for attempting to "silence" their dissent to avoid said label is not a new thing. I wouldn't be shocked if this was happening under Clinton and Reagan and others. The issue of course is that 1) I was a bit too young to truly follow those guys closely and 2) the lack of social media and news coverage made such things less likely to permeate the mainstream conciousness of the public. However, in this 21st century it's been a tactic both sides have routinely and continually used.
But there's a grand difference between fighting back against speech through the use of speech and fighting back against speech through the use of government power. The latter is closer in relation to what occured in Nazi Germany and what, if it clearly begins to occur in this country, is definitely worthy of great worry. The former, which is what this largely is today, is a sad notion of the state of politics in this country but is hardly the precusor to an authoritarian state.