Is it OK for a Catholic chaplain to participate in a “secular” prayer service and should he mention Jesus?

While I’m not sure exactly what you mean by a “secular prayer service” (since the phrase is an oxymoron), for purposes of discussion, I’ll assume that you mean a prayer given in a public setting such as a prayer breakfast or even a service where many differing religions are present such as an event marking “World AIDS Day” or the like. The fact of the matter is that Catholic chaplains, whether military or hospital or police chaplains are often asked to participate in prayer services where people of diverse religious traditions will be present. They can do so in good conscience. Obviously, the nature of the prayer will be dictated by the type of service in which it will take place. While, as Christians, we pray in the name of Jesus, we also have the practice of addressing our prayers directly to the Father. Again, the nature of the service will determine whether or not mentioning the name of Jesus will have a unifying or divisive effect.

As a practical matter, I have always found that those present at such services are very well aware that the priest chaplain is a Catholic and holds and believes all that the Catholic Church teaches and believes to be revealed by God. Most have a great respect for the Church and they expect the chaplain to exemplify what is best about the Church. That is usually why they ask him to be at the service in the first place. The mere presence of the priest chaplain is a powerful witness to the Gospel. As St. Francis is one reputed to have said, “One should always preach the Gospel, and if necessary, you may use words.”