I like to say brachos out loud at home, however when I’m at work I am more reserved. If I do make a bracha at work out loud can I instruct the non-Jews to answer amen? By a non-Jew saying amen is he acknowledging G-d’s existance? Not only is this permitted, is it recommended in terms of kiddush Hashem?
Interesting idea. Among non-Jews, the principle of stating the beracha out loud would not formally apply, because a non-Jew’s amen is not ‘part of the blessing,’ as a Jew’s amen is.
However, non-Jew’s have an obligation to believe in Hashem, and there is no reason why they shouldn’t answer amen after the blessing.
The element of kiddush Hashem depends on the nature of the non-Jews; if they are believers, and would take the idea seriously, then you can give it a try.