It is definitely legal to work outside the US if the other country allows you to work there (except if the other country is Cuba or somewhere else the US has deemed off-limits). The question is whether the USCIS will see it as a break of continuous residence. 3-4 months working for the same old US employer most probably would not be seen as such a break.
I've searched the archives trying to find any factual information about loss of continuous residency for trips less than 6 months and found no conclusive evidence. Perhaps this begs the question: if there is no presumption of break of continuous residency for travel less than 6 months, how would USCIS determine you actually worked oversees unless you told them? For example, if I decide to go on a safari for 3 months in Africa and worked as a tour guide but didn't disclose this to USCIS during my application, how would they accurately conclude if I broke continuous residency since it's not presumed I abandoned residency in the first place?
Suppose the answer is that your 3-4 months will delay your citizenship eligibility for 5 years. Would that make you decide not to go?
You must be thinking of the Reentry Permit (Advance Parole is for people who don't yet have a green card). It doesn't preserve residence for naturalization purposes. There is another form for that, the N-470, but it has strict eligibility requirements and is unnecessary for a single trip of 3-4 months (but multiple 3-4 month trips adding up to more than a year is a different matter).