The pronunciation of "et cetera" is an extremely common pet peeve, to the extent that there is a lot written about it on the Internet already. E.g.
It's one of the particular words that people tend to give as examples of "mispronunciations" that drive them nuts, or that kind of thing, along with the supposedly common "mispronunciations" of words like specific, supposedly and library. I find it a bit odd that you hadn't run into any information about this before posting your question here on this site.
You can always check a dictionary to see what it gives for the pronunciation of a word. That seems more productive than getting into a "big argument" with people who probably don't know as much as dictionary makers. For example, Merriam-Webster gives your preferred pronunciation first: "\ et-ˈse-tə-rə , -ˈse-trə also it- , nonstandard ek- , nonstandard ik- \". (But there's no way to force your coworkers to take that seriously if they are attached to their accustomed pronunciation.)
So, there is no rule saying that et cetera "should" be pronounced as "exetera". It's perfectly fine to pronounce it with /ts/ as you have been doing. The reasons for pronunciation variants are often unclear, but I've seen it suggested that the pronunciation with /k/ arose because of the influence of words starting with the common prefix ex-, which is pronounced with /ks/. There are also supposed to be pronunciation variants of escape and especially that have /ks/, and of course /ks/ is extremely common in espresso (although still stigmatized by many people).
There is no simple definition of "correct" pronunciation, so if you want to ask if a pronunciation is "correct", you have to specify what criteria you are using. (Or rather than asking about correctness, you could ask about something else that is less arguable.) There are many pronunciations that are commonly accepted, but that don't correspond to etymology or spelling; e.g. fuchsia is pronounced as /ˈfjuːʃə/ "few-sha" despite coming from the German name Fuchs [fʊks] (more or less "fooks") and being spelled (in standard written English) with "chs", which as far as I know does not correspond to the sound /ʃ/ in any other English word. Rationale is pronounced /ræʃəˈnæl/ or /ɹæʃəˈnɑːl/, despite coming from the Latin word rationale, where the e is pronounced and not silent (compare and contrast with the pronunciation of simile). Colonel is infamously pronounced kernel.