Gemara

The Mishnah there (in Yoma) says about the good appointees "Zecher Tzadik li'Vrachah" (mention of the righteous is for a blessing. This is the Gra's text.)

Yoma 37a (Beraisa - Rebbi): "Ki Shem Hash-m Ekra Havu Godel lEi'lokeinu" - Moshe told Yisrael "when I mention Hash-m's name, give greatness [to it];

Chananyah ben Achi R. Yehoshua says, "Zecher Tzadik li'Vrachah" - the Navi said, when I mention the Tzadik of the world, give a Berachah [to Him].

Rishonim

Tur (OC 192): The Mezamen answers "Baruch she'Achalnu mi'Shelo uv'Tuvo Chayinu Baruch Hu u'Varuch Shemo."

Darchei Moshe (2): The Rambam does not mention Baruch Hu u'Varuch Shemo. Many meticulous people do not say it. Perhaps the Tur holds that the Mevarech says Baruch Hu u'Varuch Shemo to separate Birkas ha'Zimun from Birkas ha'Mazon. I do not know a source for this. The Gemara, Rif and Rosh did not mention this. It seems like an extra matter. It does not apply to Birkas ha'Mazon more than to other Berachos. One could say Baruch Hu u'Varuch Shemo after any Berachah!

Bach (2): The Maharshal and others say that Baruch Hu u'Varuch Shemo is a printing mistake in the Tur. This is very difficult, for it is found in all texts of the Tur! I later saw that also the Roke'ach and Avudraham say so. We must say that they had a tradition that from their fathers and others that this is the text of the Zimun. The Tur followed them. Perhaps it is because in Berachos 45a, we learn Zimun from "Ki Shem Hash-m Ekra Havu Godel lEi'lokeinu", and in Yoma Rebbi learns from here that one who hears Hash-m's name must give greatness [to it]. Chananyah taught that when one mentions the Tzadik of the world, others must bless Him. Early Chachamim derived that regarding Birkas ha'Zimun, we must bless and aggrandize His name in two ways. One is to say Baruch she'Achalnu mi'Shelo... The other is to further give a Berachah, i.e. Baruch Hu u'Varuch Shemo, since we learn both of them from one verse. The Sifri at the beginning of Ha'azinu says so. In any case, I say that the early Chachamim established this text only for a Zimun with 10, in which we say Elokeinu, i.e. Nevarech Elokeinu she'Achalnu mi'Shelo, and they answer Baruch Elokeinu she'Achalnu mi'Shelo uv'Tuvo Chayinu Baruch Hu u'Varuch Shemo. Chazal said about this "Ki Shem Hash-m Ekra Havu Godel lEi'lokeinu". This connotes these two that we said. (Baruch she'Achalnu mi'Shelo, and Baruch Hu u'Varuch Shemo - PF.) There is no reason to say this when we make a Zimun with three. People err and say it all the time, even an individual without a Zimun!

Poskim

Shulchan Aruch (OC 124:5): On every Berachah that one hears, in every place, he says Baruch Hu u'Varuch Shemo.

Magen Avraham (9): Surely, if he is in a place where one may not interrupt, it is forbidden to say this.

Dagul me'Revavah: All the more so, one may not say it in a Berachah in which the Shali'ach Tzibur is Motzi him, e.g. the Shofar or Megilah.

Gra (DH Al): This is like it says in Yoma. See the Tur.

Kaf ha'Chayim (25): Divrei David (41) says that one who is learning should not interrupt his learning to answer Baruch Hu u'Varuch Shemo or Amen, and so says Mayim Rabim 2. Pischei Teshuvah disagreed, but said that one may rely on them for Baruch Hu u'Varuch Shemo. However, for Amen, what is their source? The Gemara obligates answering Amen! One who learns all the time, like R. Shimon bar Yochai and his colleagues, who are exempt from Tefilah, or one who teaches b'Rabim, who is exempt from Keri'as Shema (they are exempt from answering Amen).

Mishnah Berurah (21): If he is in a place where one may not interrupt, e.g. Pesukei d'Zimra, and all the more so Birkos Keri'as Shema, even Bein ha'Perakim, one may not say this. Similarly, if he heard a Berachah that he is obligated to say it, and he intends to be Yotzei through the one saying it, e.g. Shofar, Megilah, Kidush, or similar matters, he should not answer Baruch Hu u'Varuch Shemo, for Shome'a k'Oneh. It is as if he said the Berachah himself, and it is a Hefsek. Chayei Adam was unsure about [whether he was Yotzei] b'Di'eved. I say that b'Di'eved, one should not be stringent. See what Magen Giborim brings from Ma'aseh Roke'ach. (See Kaf ha'Chayim below.)

Kaf ha'Chayim (26): Devar Shmuel says that one who is obligated to say a Berachah may not answer Baruch Hu u'Varuch Shemo. Since Shome'a k'Oneh, it is a Hefsek. It is as if he changed and added to the text of the Berachah. Ma'aseh Roke'ach (Hilchos Berachos 1:11) disagreed, and supported himself from the custom to answer Baruch Hu u'Varuch Shemo in Berachos of Shofar and Lulav, even though people are Yotzei through the Shali'ach Tzibur. Shoshanim l'David (Berachos 8:8) refuted all his proofs, and supported Devar Shmuel. Mateh Yehudah (or Mateh Yosef?) agreed, and so did Dagul me'Revavah and Kitzur Shulchan Aruch. The Birkei Yosef said that it is proper to be careful not to say it, but we do not protest against one who says it. Zechor l'Avraham (or Zechor l'Yitzchak) agrees.

Mishnah Berurah (22): A hint to Baruch Hu u'Varuch Shemo is "Ki Shem Hash-m Ekra Havu Godel lEi'lokeinu." Also, if when one mentions the name of a mortal [Tzadik], he must bless him, for it says "Zecher Tzadik li'Vrachah", all the more so one must do so when he mentions the Tzadik of the world. Berachos that have a short ending of only two words, e.g. Poke'ach Ivrim, in Birkas ha'Shachar and similarly in Shemoneh Esre and all Berachos, it is proper for the Shali'ach Tzibur to be careful not to end it immediately, rather, to wait a little, in order that all those who answer will hear which Berachah he finishes afterwards, and they will be able to answer Amen immediately. If he does not wait, sometimes due to the sound of people answering Baruch Hu u'Varuch Shemo, many answerers do not know which Berachah he finishes, and their answer is not immediately after the Berachah finishes, and there is concerned for an orphaned Amen (Rema, Sa'if 8).

Kaf ha'Chayim (27): One answers Baruch Hu u'Varuch Shemo also when Kohanim say Hash-m's name in Birkas Kohanim. Shirei Keneses ha'Gedolah, the Pri Chodosh, Birkei Yosef, Kaf ha'Chayim (Palagi) and Ben Ish Chai say so. Shalmei Tzibur says that Devar Shmuel connotes that one should not say it, but the custom is to say it.

Kaf ha'Chayim (28): If you know that the Shali'ach Tzibur will rush and finish the Berachah while you say Baruch Hu u'Varuch Shemo, do not say it, so you can answer Amen immediately, without say that Baruch Hu u'Varuch Shemo Amen without a break in between. Amen is more important Baruch Hu u'Varuch Shemo, for it all depends on the closing. Amen includes Baruch Hu u'Varuch Shemo (Yefeh l'Lev).

Kaf ha'Chayim (5:2): When saying Baruch Hu u'Varuch Shemo, one thinks that Hash-m is blessed by Himself, and does not need our Berachah.

Shulchan Aruch (192:1): The Mezamen answers "Baruch she'Achalnu mi'Shelo uv'Tuvo Chayinu Baruch Atah Hash-m..."

Taz (1): The Bach says that we say Baruch Hu u'Varuch Shemo only when there is a Zimun with 10, but not when there are only three. This is difficult. If so, the Tana should have mentioned this distinction in the Mishnah! Therefore, I say that there is no source for this from the Gemara and Poskim.

Mishbetzos Zahav: The Bach concludes that Godel ("Ki Shem Hash-m Ekra Havu Godel lEi'lokeinu") applies when there are 10. Then, those who answer must say "Godel", after answering Baruch she'Achalnu, and add Baruch Hu u'Varuch Shemo. The Taz asked why the Mishnah did not mention the distinction between three and 10. I can answer for the Bach that Rav Asi said "Gadlu la'Shem Iti u'Nromemah Shemo Yachdav" applies to a Zimun of three. R. Avahu says that "Ki Shem Hash-m Ekra Havu Godel" applies to 10. If so, we must say that there are words missing from the Bach. Based on this, the answerers say Baruch Elokeinu she'Achalnu mi'Shelo, and also Baruch Hu u'Varuch Shemo, which is the addition. However, the Tur did not mention this. He said only that the Mevarech says Baruch Hu u'Varuch Shemo. Darchei Moshe says that the Mevarech says Baruch Hu u'Varuch Shemo to separate Birkas ha'Zimun from Birkas ha'Mazon. If so, he says it, but those who answer do not, for they separate through answering to the Mevarech, like the Shlah wrote.

Magen Avraham (Reish ha'Siman): The Shulchan Aruch connotes that we do not say Baruch Hu u'Varuch Shemo. Mateh Moshe says so in the name of Maharshal, and the Bach says that we say it only with 10. The Tur, Avudraham, Maharal and Levush say to say it, and this is the custom. Surely an individual does not say it.

Mishnah Berurah (4): This connotes that one does not say Baruch Hu u'Varuch Shemo. Some Poskim say to say it. In any case, it does not apply to an individual. Also when there are three, only the Mevarech says it, but not the two who answer.

Kaf ha'Chayim (8): The Shulchan Aruch did not mention Baruch Hu u'Varuch Shemo here, nor the Rema. Since there is an argument among the Poskim, and it is not a Chiyuv, it is best to be passive. However, where there is a custom, we do not protest.