Here we go again. I've redone the family tree now three times. I'm surprised any Gillon can trace back farther than 1700. The Torphichen christenings are missing between 1692 and 1697 and 1712 and 1714. Torphichen marriages are missing between 1714 - 1808. Bathgate christenings are missing between 1684 and 1722 and no marriages in Linlithgow between 1719 and 1730. However, I set myself a spreadsheet showing the various Gillon families between 1600 and 1777 when my John Gillon married Margaret Anderson, and filled in where the blank records would be. John christened his eldest son Andrew which would indicate his father's name was Andrew. Andrew Gillon and Agnes Schaw were married in Bathgate 4 Nov 1732 but moved to Torphichen. The christening record of their son, John 26 Oct 1735 in Torphichen says, "To Andrew Gillon and Agnes Schaw in Wallhouse Loanend (?), a son called John. Witnesses John Gillon and John Lermouth." The christening record of their daughter Janet 18 Nov 1733 says, "To Andrew Gillon and Agnes Schaw in Wallhouse Loanen, a daughter called Jannet witnesses John Gillon and Jas. Walker." For some reason, by the time they had William christened 16 Apr 1738 in Torphichen, they had moved. "To Andrew Gillon and Agnes Schaw in North Alderton (Adderton ?) a son called William witnesses John Mitchell and William Thomson." Now was Andrew Gillon another son of Alexander of Wallhouse born between 1712 - 1714? Or was he another illegitimate son? My other concern with this being the father of my John is that my John Gillon would be forty-two when he married Margaret Anderson.
If John (Andrew) is not my John Gillon then the only other possibility from Bathgate is that of Alexander Gillon the younger of Wallhouse and Elizabeth Muir. They had three recorded children. Margaret christened 21 Apr 1739 in St. Cuthbert's, Edinburgh; Ann christened 5 Jul 1741 in Bathgate, and Alexander christened 15 Sep 1745 in Bathgate. From 1745 until the 1760's not only was the country in a turmoil but the Presbyterian church had a couple of breakaway sects which disrupted the parishes of Torphichen, Linlithgow and Bathgate. No doubt records have been lost or not kept and there is some doubt whether Alexander actually married Elizabeth Muir. If John Gillon was the product of this arrangement, his son William might very well have changed his name to Anderson, feeling perhaps that this name was not his legally.
The bottom line...I can't prove where John Gillon came from other than "Bathgate".