1. Jeremiah (b.1720s?)
2. William (died 1793)
3. Nathaniel (died 1810)
4. Joseph
5. Judith

Also see: The first 2 generations in America

¹much of the info on this line provided by Ann J. Everingham.

Thomas, and his brother fought in the Revolutionary War against the british.

Info from Bill Malkin, 2000:
"Have pondered over the info you sent and do think that the Thomas Everingham and wife Jane who deeded land to George Hulit on April 1, 1803 is the same Thomas Evernham of East Windsor, Middlesex County whose estate was inventoried on Dec 8, 1810. (Calendar of Wills 1810-1813). This inventory includes obligation against George Hulit. there may be more papers concerning the "obligation" in Middlesex County to widow Jane Everingham or perhaps to Jane McCabe. We know that James McCabe and his wife Jane were both alive on 20 Feb 1819 when James McCabe wrote to Zedekiah South and speaks of "your step-mother and I" In the letter from James McCabe he mentions your step-mother and her son, William Everingham and his brother, John who moved to Genessee County, NY".

⁴SPECULATION: (Tom Phillips, 2006) No absolute proof but it is believed that Thomas (link above) is the son of Thomas & Jane. Records have indicated that Thomas was known as Thomas Everingham Junior. While this does not definitively tie him into the H&R1 Family line the other coincidences (i.e geographic location, 1st daughter named Jane, 1st son was named ASA; probably after ASA Kitchell father of Tryphena who married Enoch Everingham (FL6). I can convince most observers that Thomas was the son of Thomas & Jane Hulit Everingham and another grandson of Henry & Rebecca Everingham.
This would mean that Thomas Everingham & Jane Hulit had more than the two sons (William & John "who went to Genneessee County NY" ) that were listed in a letter dated 1819 from Jane's new husband, James McCabe. Jane Hulit Evernham McCabe ended up living right next door to George & Lucy Evernham according to the 1840 Census for Mercer County NJ.