The Shulhan Aruch (Orah Haim 182) writes that Birkat Ha’mazon is Halachically equivalent to the Amida prayer, the only difference being that the Amida is reciting standing, and Birkat Ha’mazon is recited sitting. And thus, as the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) rules, one should not interrupt Birkat Ha’mazon to answer "Amen" to a Beracha, or even to respond to Kaddish, Kedusha or Barechu. However, the Ben Ish Hai adds that if one hears a congregation reciting "Modim" as he recites Birkat Ha’mazon, he should silently nod his head as the congregation bows. If he hears Kaddish, he may silently listen, but he should not answer.
The Ben Ish Hai writes that these laws apply from the beginning of Birkat Ha’mazon through the end of the fourth Beracha. After the four Beracha, during the "Ha’rahaman" section, one should remain seated and recite the text with proper concentration through the very end of Birkat Ha’mazon, but one may answer "Amen" and respond to Kaddish, Kedusha and Barechu while reciting the "Ha’rahaman" section. (Technically speaking, one may converse in Torah matters during the "Ha’rahaman" section, but this should not be done in front of ignorant people who might then conclude that speaking is allowed during Birkat Ha’mazon.)
The Arizal (Rav Yishak Luria of Safed, 1534-1572) taught that ideally, one should recite Birkat Ha’mazon with his eyes closed, and with his right hand over his left hand with the thumbs tucked in the hands. Of course, if one does not know Birkat Ha’mazon by heart, then he should read it from a Siddur, and hold the Siddur if necessary. And, if one recites Birkat Ha’mazon over a cup of wine, he should hold the cup and look at it during Birkat Ha’mazon.
One must sit during Birkat Ha’mazon; it should not be recited while standing or while reclining. The Shulhan Aruch brings opinions that the Beracha of Me’en Shalosh should likewise be recited while sitting, and this is also the opinion of the Ben Ish Hai and of Hacham Ovadia Yosef, in Yabia Omer (vol. 1, Orah Haim 11:14).
Numerous books emphasize the great importance of reciting Birkat Ha’mazon with concentration. The Kav Ha’yashar (Rabbi Zvi Hirsch Kaidanover, 1648-1712) writes that one who recites Birkat Ha’mazon with concentration will earn the privilege of hearing King David recite Birkat Ha’mazon at the meal of the Sadikim. And the Arizal, in Sha’ar Ha’misvot (listen to audio recording for precise citation), writes that one who recites Birkat Ha’mazon with concentration receives his livelihood joyfully and comfortably.
Summary: While reciting the four Berachot of Birkat Ha’mazon, one may not make any interruptions, even to answer "Amen" or to respond to Kaddish, Kedusha or Barechu. During the "Ha’rahaman" section, one may interrupt for these responses. Birkat Ha’mazon should be recited sitting and with concentration, and one who does not need a Siddur and is not holding a cup of wine during Birkat Ha’mazon should recite the text with his eyes closed and with his right hand over his left hand, and the thumbs tucked into the hands.
