The one thing you almost certainly could not do in these cases is any sort of DNA study. DNA has proven to be a wonderful molecule on which to base life here on Earth but there’s no reason to think that an alien life form would need it.
In fact, if we found alien life that was based on DNA, then that might lend some support for some sort of panspermia model for life on Earth. In other words, life started elsewhere and was seeded here on Earth (and possibly lots of other places too).
What you need for life is something that can make copies of itself almost perfectly*, is stable, and can carry a message. There are probably many different chemicals that fit this bill.
Unfortunately, without knowing something about this alternative genetic material, we probably could not make any sense of it even if it was hiding in some alien bone. Maybe we’d find some sort of chemical that was similar to DNA in that it had lots of repeating units. But we wouldn’t be able to read it. And we’d be hard pressed to figure out how it works without something living to put it into.
Of course, this assumes that whatever molecule stores the alien’s information is stable enough to be found. If the bone was very old, the molecule would have to be much more stable than DNA.
Scientists have gotten very good of late at reading the DNA of long dead creatures. They’ve even managed to piece together all of the DNA of a Neanderthal who lived tens of thousands of years ago. But they can do this because they know how DNA works and can exploit that knowledge to make lots of DNA from the scraps left behind.
Without knowing how the alien storage molecule works, we couldn’t make more of it. Which means there needs to be a whole lot there for us to even get started on understanding it.
What all of this comes down to is that in most cases it would be impossible to use any sort of genetic analysis to prove something was of alien origin. Except for a few rare cases, you’d need to rely on where the bone was found or what the intact skeleton looked like or if it was clutching some bit of advanced technology. The best proof would probably be meeting one in person…
*The almost is key here. You need for there to be some error rate so that evolution and natural selection can happen.
37.7749295 -122.4194155