Paying respects during grace, benediction, etc. as a non-Christian? I'll make this clear, I'm not Christian, I don't have any religion.  But I respect all forms of religion regardless of what the beliefs may entail.   As a Boy Scout in my younger days I was taught to be reverent and to respect all religions, so I've been curious about this particular quandary I've been in for a bit.

During grace, or benediction, when the speaker makes a prayer (I assume that is what it is) and everyone else is respectingly bowing their heads and closing their eyes (I think this is the right way, correct me if I am wrong), am I in the wrong if I do as everyone else does?  Meaning, as a non-Christian, am I offending Christians if I do what they are doing even if I am not Christian? Is there a way that's okay for me to do as to not disrespect others?
While there may be some differences in how some particular Christian groups respond in such a situation, it's safe to say that the vast majority will not be offended by what you're doing. Bowing your head is certainly a sign of respect - primarily to God, but also to the community you are with. In most cases it would be neither here nor there if you close your eyes  or not - perhaps whatever you find most comfortable (when people do this, it's usually just to help them focus rather following a rule to do so). 

One part of things that could potentially be more of an issue is where there are "call and response" type prayers - a prayer leader initiates and the congregation responds, whether merely with "Amen" (a very common response to "Grace" in a corporate setting) or some other liturgically required response (as in some benedictions). In this case, it is probably preferable that you only respond with what you yourself agree to. "Amen" actually means "Let it be so", so if you don't agree with what was just prayed, then rather than just echoing it with everyone else, a respectful silence would avoid hypocrisy.