negotiating out of state tuition for public university

That would depend entirely on the public university system and you should e-mail their guidance counselors or staff to see whether you can.

If in doubt, try using programs like collegevine to see what exactly you can negotiate on.

You can definitely try it, but it's pretty unlikely as an OOS student to be honest. Still, it depends on the school, the program, and how strong you are as a student. If they've already given you money and you want more, it may be more likely than if they've given you nothing and you're asking for some. It also depends on the leverage you have—what other offers can you show them that would be better/cheaper? Schools use merit aid to recruit people. If you don't have anything like that, it's really unlikely that you'd get anything from them, because they know they wouldn't need to give you more money to compete with anywhere else.