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そのときに歴史の先生がこう呟きながら連れて行かれます。
Source: https://www.youtube.com/watch?v=3xs-XWk_oQA 1:17
I'm assuming it's the word 斯う meaning "in this way," and here it would be describing how the teacher said the quote to follow. Is this こう anything like the quotative like in English that can be used for either literal quotes or paraphrasing? For example, "He was like, 'When is history not essential?'"
Yes, your understanding about こう (I've never seen it written as 斯う though) meaning "like this" is correct, however in some cases it can be more thought of as "as follows" or "as below".
Here is a simple example:
そして、僕はこう言った。 [And then, I spoke.]
「日本語なんて難しすぎるよ」 [Japanese is just too difficult!]
As in this example, often it's tricky to translate since saying "I spoke this" sounds unnatural, but you'll be fine if you understand it can be used to refer to something that follows immediately.
Another example:
「僕」という字はこう書くよ [This is how you write the character ”僕”]
In this case, the speaker is likely writing the character as they speak.
「こう」is the adverbial form of the こ-series（これ、この、ここ、ect.). A close literal match in English is 'thusly' (in this way).
