Let's quote the advice in its entirety rather than a snippet.
Inverted word-order in non-questions: After negative elements
The same inverted word-order as that found in yes/no questions is also
used in clauses introduced by a negative or restrictive clause
element. In the following example, the initial element Not only is
negative and is followed by inverted word order: could congress
(12) Not only could (v) Congress (S) declare (V) war but the states were forbidden to engage in it without the consent of Congress.
Further examples (with the initial negative element highlighted):
(13) Never before have human rights been so fully and completely violated.
(14) On no account must the moisture level raise above 7 or 8 per cent.
(15) Not until the end of the hour-long conversation did the President get to the point.
“Is the sentence Not all my cigs lit a candidate for inversion in the first place? And why or why not?”
The answer is yes. The reason being that any sentence can be inverted. Why couldn't it be?
But … before we get to a demonstration with your sentence let's choose a different sentence because when you put lit in front of cigs it can function as an adjective and that's confusing the issue. (The issue is further confused because I can't tell if you actually want to start out with the sentence My cigs lit or even All my cigs lit rather than Not all my cigs lit.
Take Not all the birds played tennis.
Now clearly you can't say * Not all the played birds tennis.
That is never a construction in English. Similarly Not all my lit cigs could never be a construction in English if lit is a verb.
You will notice that all the inversions in the example use an auxiliary verb, as in the could before the small (v). Namely could, have, must, did.
In fact, the section before the one you quoted from your link goes into that.
You could invert your sentence like so: A tragedy! Never before this ill-fated day had (v) all my cigs (S) not lit (V). Note the not moved.
Incidentally. Not all is neither a negative nor restrictive clause element. It's a quantifier. It gives an indication of quantity. How many? Not all, that's how many. If it is your intention to invert All my cigs lit or simply My cigs lit then you might end up with something like Not until I found my zippo did (all) my cigs light. (The tense has not changed, it's a mere trick of the light.) The Robert Frost quote
Not all your light tongues talking aloud / could be profound
is not apposite because light is being used here as an adjective -- not a verb -- and not all is, as we have seen, a quantifier.
IANAGM (I am not a grammar monkey) -- Usual health warnings apply.