Here's a possible match for your Green Arnold from 1870 Collin Co, TX.
1870 United States Federal Census Precinct 1, Collin Co, Texas John Broughten 20 m w chair maker $350r $1200p MS md in April Lydia Broughten 32 f w keeping house (born MS? or MO?) md in April Eliza Howard 55 f w at home AL John Howard 14 m w at school TX Isaac Howard 13 m w at school TX Elisabeth Howard 11 f w at home TX Jane Howard 8 f w TX Greene Arnold 10 m w at school TX
Since the 1870 census doesn't list relationships it's hard to know how the members of the household are related. According to information submitted by an LDS member, John Broughten married Lydia Crabtree in April 1870 in Collin Co, TX. So Lydia is probably not Greene's mother. I guess Eliza Howard could be John Broughten's mother and the younger Howards are her children from a second marriage. But none of that helps to know if Greene Arnold is related to them or not.