A Jewish man is sitting at his desk at work. He just said “Baruch ata Hashem Elokeinu Melech” of Birkas Hamazon when his gentile boss walks in, accompanied by an entourage of distinguished gentiles in the company, who desire to greet him and ask him certain questions. The head of the entourage, a non-Jewish lady, introduces herself and extends her hand to the Jew while he is in the middle of Birkas Hamazon. What should he do? He may not speak nor may he shake her hand. He is bentching by heart, without having a siddur in front of him, so he cannot even point to his siddur so as to indicate that he is in the middle of bentching.
You are portraying an awkward situation. First of all if he is bentching with kavana, they will all see that he is praying, and they will respect him for that. He can also say the words louder so they will understand that he is doing something religious. in this case he would also be allowed to motion to them that he will be with them shortly, because he is doing it in order to help himself concentrate better on his bentching. After he is finished he can politely explain that he was praying, and thanking G-d for the food that he just ate, which is one of our commandments. Other than that we have to know that there are times that we will have nisyonos, and we will be rewarded for it… and it might also cause them to respect him for it more.
I know of a story of a man that was in negotiations with a big company in Japan over a million dollar deal. When he got to Japan he met the vice president of the company, a woman, who extended her hand to him, and he politely explained to her that it is against his religion to have physical contact with any woman other than his wife. after the meeting he was told that the company will give him an answer in two weeks. He thought, “That it after all of that I blew it” and he was all dejected. Two weeks later he got an answer from the company that they want to double the original order. Their reason was that if this man can stand up to his ideals even in an awkward situation, that means that he is a man of principles and values, and therefore they want to do business with him!!: