We don't know much about Roger Lillis or his son, John Walter, including a confirmation of their names! Nor have I even located the family in a census, yet. Our info is that John was married in 1892--but we don't know where. Their daughter Abigail (my grandmother) was born in Dodge City, KS, in 1896. In 1917, she marries Lindsay Kirkham and settles in Independence, MO (just outside Kansas City). Kirkham's father and grandfather had been in Wyandotte, KS, as far back as 1860. Since we have matching surnames, the same county in Ireland, and settlement in the same state, I'm also wondering if Roger and Thomas might be brothers. Have you found the ship Thomas came over on? That might mention siblings or others who traveled with him.
On the other hand, Lillis may simply be the most common family name in County Clare, so that among emigrants to the US named Lillis, all or most originated in Clare but may nevertheless be unrelated to one another.
The 1870 Census for Wyandotte County lists a LILLIS, Patrick, Page 598, Ward 1 of Wyandotte City. So far, I see no connection between Patrick and yours or my families -- but it sure is odd that so many Irish folks of that surname were hanging around the same area of KS at that time.
I'll post whatever new information I find. I think my next task is to find birth and marriage records for some of these folks, to get the facts down. Keep in touch!