R. L. Billington's sister Ella was my wife's gr-grandmother. Thanks to folks like you it was easy to trace back to Ezekiel. Beyond Ezekiel, things get nebulous. Here's the information I have on Ezekiel and some sources.
His birthdate appears on his Revolutionary War pension record No. W8365. In his interesting narrative, he tells of how he served in the militia of Essex County, NJ. After his capture and escape from the British, he fought at Eutaw Springs and went to NC after the war where he married Elizabeth Penny. He later moved to Tennessee, where you can find information in Bedford, Marshall and Maury Counties on he and son James of TN.
Back on the Mayflower side of Ezekiel's ancestry, the New Jersey archives list James Billington of NJ as early as 1747. New Jersey colonial wills have him as witness on several. Since James Billington was the only Billington in Essex County, NJ and since he was married to Anna America (Garner?) at the time Ezekiel was born, James probably fathered Ezekiel. The only Nathaniel Billington that I was able to find during that era lived in MA and was born ca 1750, which would make him too young to be Ezekiel's father. So far, the evidence overwhelmingly supports James as Dad. I know of no documents where Ezekiel would have listed James as his father.
Let me know how you searches turn out! Please email me if you would like to share information.