I've listened to him an awful lot. Not sure what to make of it, but it is quite a good story. I'm not aware that he has ever offered anything other than his own words to prove it. In the end, I enjoy it just the same.
If I recall correctly, Stone has said that he never took an oath of secrecy about his work with aliens. I don't remember him ever reporting any attempts to intimidate him into silence, visits from MIB, or anything like that. So, if he is telling the truth, perhaps the military has deliberately permitted him to run with it as a form of soft or slow disclosure.
One thing I wonder about is his health problems. If he really is in contact as he says, why not ask them to help him overcome those health issues. That would go a long way to proving his stories.