No I meant to say Church of Christ which has no denomination.
Your definition of "denomination" may need a little tweeking. The word was coined in 1398 from Latin de- + nominare, and literally means "to assign a distinctive name". A denomination is "a group of religious congregations having its own organization and a distinctive faith" and can also be described as "a group within the Christian church". A person can only be called a non-denominational Christian when he does not worship in an organised group.
They have their own special name; they don't merely call themselves "Christians"... you cannot write a cheque to "Christians" and have it sent to your intended recipiant. You can however write a cheque to "the Watchtower Society" or to the "Church of Christ". The very fact that you said "There's a distinction between the UCC and the Church of Christ" shows it is separate from that branch of Christianity. They have "assigned themselves a name" and are therefore a denomination.
finding, i definately recommend you follow jgnat's advice and take things slow with this religion. Even though they may not force you to be formally baptised, they still have distinctive beliefs, and as you learn more and more about those beliefs, you may find they conflict with your view of what Christianity is or should be. Never fall into a trap of wanting your whole family to worship in the same way, it may cause unnecessary conflict espacially if you discourage your family from expressing their own individuality. Maybe not as grave as if you were in the Watchtower, but damaging nonetheless.