Han is CG. Right from the beginning.
Shooting Greedo first doesn't make Han nuetral. It makes him sane. I wouldn't even object to lawful good shooting first under those circumstances. Clearly Han is under lethal threat already, and Greedo is no saint, not even a just magistrate of a just society, not even an honorable foe for whom a chivilrous person would be obligated to deal with honorably.
Although they aren't cannonical, Han's CG nature comes out even more strongly in the old trilogy of prequels about Han's youth.
Han is jaded. Han is cynical. Han is a little world weary by the time he meets Luke, but that is just his personality, not his alignment. Han is a smuggler, but for the most part - at least until he gets work from Jabba - Han is breaking laws which are unjust and the product of an unjust tyranny. He is no more evil for doing so than Leia and the rebellion is evil (or even Chaotic!) for breaking the Empires laws. From the standpoint of even a lawful, the Empire could have forfieted its right to govern by breaking the meta-laws which established its right to rule. Of course, Han shows his inability to suffer the Empire by going AWOL from the Academy, hanging out in the underworld, rescuing Wookies from slavery and otherwise acting the part of a scoundrel - except when a real test of good or evil is on the line, and Leia and the rest do the whole Republic in exile thing, but what at stake in that is more a matter of chaotic versus lawful than good versus evil.