A quick browse of “The Upper Room,” the website for “Walk to Emmaus,” reveals that the organization affiliates itself with The United Methodist Church. The incredibly limited program overview mentions daily Holy Communion (which, as a Catholic, you obviously could not participate in) and “a 72-hour short course in Christianity” comprised of faith talks on the meaning of Christianity and what it is to be the church. Such lectures and discussions are surely filled with Protestant teaching which, if you are not expertly versed in Catholic teaching, may be problematic for you.
Perhaps the most confounding thing about the “Walk to Emmaus” program is that in their bookstore, The Upper Room charges for a copy of what they call “What Is Emmaus?” a booklet of information about what happens during an Emmaus weekend, what to expect, and FAQs. This would lead me to believe that there is possibly something illegitimate going on with the program as no retreat should ever charge prospective participants for information on what goes on during the program.
I think that you should politely decline the offer. You will benefit far more from a Catholic retreat where you can participate in daily Mass, hear talks about orthodox Catholicism, and share your faith with other Catholics.
There is a wonderful retreat program put out by a Catholic organization which happens to be called Emmaus as well. I know several people, including my parents, who attended the retreat and had a marvelous time. You can find more information on it here.