So, to revise in light of more recent evidence, Utsuro is white-haired and powered by the Altana. The white-furred Sadaharu's mystical powers as Altana guardian proves that Utsuro isn't unique in this ability. Oboro has white hair and his master's ability of meridian manipulation, along with borderline immortality, making a good case that he too is an Altana user.
Between Gin's insane endurance, regeneration and ability to grow stronger the more wounded he is, white hair, demon physicals, and strange meridian resistance, that is more than enough evidence for me to assume the secret to his powers is a similar connection to the Altana. Further strengthened by Utsuro drawing such paralells between them in the flashback. But since Gin is still so young it is too soon to tell if he is immortal as well.
Jirocho has only the white hair and demonic fighting prowess at an age where he was assumed to be too old for such things to indicate an Altana connection, but that is more than enough for me. Which in light of his similarity to Gin as a youth is plenty indication to assume he is the dad. A human dad, but one touched by the Altana, a trait passed on to Gintoki. Clearly not immortal, though, suggesting that there are degrees of Altana power, if I am correct. Well, Oboro complaining about almost dying would cement that fact anyway, since Utsuro simply can't.
My only problem is how people like Okita and Nobume fit in. As a former Naraku, Nobume has every possibility to be Altana-enhanced, but she lacks the white hair. Okita has no excuse for being able to cut a building apart with his sword, and regenerate from beatings as bad as Gintoki's in such short time. On its face, these characters suggest this is just a manga universe where people have the Charles Atlas Superpower, but the degree to which the white-haired ones appear superhuman even by this standard, the "Altana-powered Jirocho as Gintoki's dad" explanation makes too much sense for me to ignore.
Takasugi being down for the count for so long because of wounds Gin would just shrug off in a chapter or two is such a powerful illustration of the difference between characters in this manga. He truly is just a very skilled human in a world of aliens and Altana-powered monsters.