Jackson is a pseudonym used by Greg.
Although "Jackson" was never Greg's legal last name, it was his mother's maiden name. Greg lived most of his life with his father's legal name, Daniel, and used it often on his early internet websites, most notably his 2005 GregoryJDaniel.com website. He has since legally changed his last name with Taylor after their marriage, but their current legal name will not be mentioned on this site.
During most of his YouTube career, Greg has gone by "Gregory Jackson".
When Greg is asked about his real name, he will usually say something like "[My real name is] Gregory Jackson (according to my Mom)". (Source)
Greg would often use the name "Jackson" as a source of encouragement and strength during his military days, which can be seen in his 2005 online journal:
"I realize many cry and complain for being so emotionally abused in boot camp, yet I am convinced no such tears will reach my eyes like the others often have ~ I am full of Jackson strength, and devotion to my cause, you can all have faith in that."
"Deep down, I am not just a Jackson, I am the son of a great woman, she is the true Jackson I see, that and her father ~ the two people I respect most, one because I know her well, I've seen so many honorable things... and two because the one I respect looks up to another, and no doubt, if the one I look up to looks up to another, than I can easily call such a person also my superior, my idol." (Source)