Is it OK to use the non-Jewish system of counting years, or must we use specifically the Jewish year?
The Jewish year is based on the Torah’s reckoning of the passage of years from Creation. Using this system of counting the years thus recognizes the existence of G‑d, the fact that He created everything, that He dictates and regulates the natural world and that He guides history.
Aside from this theological consideration, even if only for reasons of seniority, the Hebrew calendar should take precedence. The current Jewish year of 5777, corresponding to the non-Jewish year 2017, makes the Hebrew calendar 3760 years older than the Gregorian calendar!
But there’s more to it than that.
The non-Jewish year reckons the number of years since the purported birth of Jesus. This reckoning was adopted in order to assert the idea that Jesus was G‑d embodied as the Savior of Mankind. This is the meaning behind the terminology B.C. and A.D. The former stands for “Before Christ” in English (christos means savior in Greek), and A.D. stands for Anno Domini in Latin which is an abbreviated reference to the phrase, “The year of Lord Jesus Christ”. (Contrary to popular mis-conception, A.D. does not stand for “After Death”, since this system revolves around the birth of Jesus, not his death.)
Insofar as using this system of reckoning years may be a tacit acceptance of the religious beliefs of Christianity which lie therein and which Judaism does not accept, a Jew should strive to use the Hebrew calendar and not the Gregorian one.
That being said, since the Gregorian calendar has become the universal standard such that it is nearly impossible to avoid using it, one may do so with this utilitarian purpose in mind. This is particularly so nowadays when these dates are applied to most spheres of life (finance, education, travel, etc.) where this system is used in a way which is void of its religious content.
However, even when using this non-Jewish way of reckoning years, where applicable and relevant, such as when citing an historical event, one should avoid the religious connotation of B.C. and A.D. by using the more neutral terms of BCE (Before the Common Era) and CE (Common Era), respectively.
In a similar vein, when noting events of a particularly Jewish nature, such as the dates in invitations to weddings or Bar Mitzva celebrations, one should use only the Jewish date. If this might cause confusion among invitees who are unfamiliar with the Jewish date, then the non-Jewish date may be used, but only in a way which is secondary to the Jewish date.