I am Jewish, live in Israel, and will soon visit my non-Jewish family in Belgium for a secular family celebration. Being a religious woman in Israel, I never shake hands with men. But in Belgium it is part of the "etiquette" to shake hands with men. If I won’t take the offered hand, it will for sure embarrass the uncles, brothers, old friends... Of course I will not offer my hand, but may I shake hands with those non-Jewish men - if they offer theirs, in order not to embarrass them?
Shalom, A woman should not touch any man other than her husband. Therefore you should make every effort not to shake hands with men, and this itself is an important educational statement about the sanctity of the Jewish people and the way that we view the relationship between men and women. However, in a case where there is no choice and it will be damaging then you may weakly shake a man's hand if it is offered as this has no sexual overtones, but this should only be in cases where there is no choice. Kol Tuv,