I have noticed in Chabad synagogues and sefarim, in the Kaddish after "sh'mei d'kudsha brich hu," "amen" is said by the congregation. In every other source I've seen and every other synagogue I've been to, "brich hu" is repeated by the congregation. Which is proper for someone who is not nusach Chabad, and why does Chabad do it their way?
The Shulchan Aruch OC 56:2 says that one says "Amen" after Berich Hu, but the Rema there disagrees and says not to say anything there at all.
The Ashkenazim who say Berich Hu at the same time as the Chazzan are following the Taz and the Magen Avraham. Strangely enough, the Shulchan Aruch HaRav also says the same thing. (I haven't seen anyone who follows the Rema over the Taz and the Magen Avraham.)
Sephardim follow the Mechaber because he's the Mechaber (It helps that the Ari Z"L agrees), but when Chabad says Amen there, they're probably doing it specifically because that's what the Ari Z"L says to do in Sha'ar HaKavanot.
- Chabad: Say "Amen"
- Sephardim: Say "Amen"
- Ashkenazim: Say "Berich hu"
Note that I'm not addressing the question of what to do if you're a non-Chabadnik who happens to be davening at a Chabad shul for some reason.
My source for all of this: Halacha Berura 56:21, and the footnotes there.
According to the Aruch HaShulchan (OH 56:6) this is really fundamentally a grammatical question.
Does the sentence run on from before: "may it be lauded the name of the Holy One Blessed is He" in which case the response is "amen".
Or does the sentence end with "may it be lauded His holy name" and then the next sentence starts with "b'rich hu": "may he be blessed above all blessings" in which case the congregation joins the start of the next sentence by saying "b'rich hu"
וכתב רבינו הרמ"א שלא יפסיק בין "הוא" ובין "לעילא מן כל ברכתא", עיין שם. דסבירא ליה ד"בריך הוא" קאי על למטה, כלומר: "בריך הוא לעילא מן כל ברכתא". אבל השיגו על זה ד"בריך הוא" קאי אדלעיל "קודשא בריך היא", כמו שבלשון הקודש אומרים "הקדוש ברוך הוא". וכן מבואר מהרמב"ם, שכתב "אמן" אחר "בריך הוא". וכן המנהג פשוט. ו"לעילא" הוא קאי על "יתברך וישתבח", שיתברך למעלה מכל הברכות והשירות והתשבחות שנאמרו מעולם.