I have found a total of zero substantiated pieces of evidence to back up a case against his ideas. In any case, he has had a long career and gotten involved in research into many controversies - it's unlikely that someone who has done this will say nothing that any single person will disagree with. That being said, I have looked into a lot of the topics he has covered before (without knowing who he was/is) and I agree with a lot of his conclusions. I haven't personally looked into the Zika virus at all, but I have looked into Ebola's origins and Lynton Bionetics was heavily implicated.

The website 'Wikispooks' typically shreds the reputation of most controversial people, but in his case it says virtually nothing that is critical (from what I read, which was not 100% of it):

Francis Anthony Boyle is a professor of international law at the University of Illinois College of Law.[1] Professor Boyle was responsible for drafting the Biological Weapons Anti-Terrorism Act of 1989, the American implementing legislation for the 1972 Biological Weapons Convention. He served on the Board of Directors of Amnesty International (1988-1992), and represented Bosnia and Herzegovina at the World Court. In the early 1990s, Boyle represented Colonel Gaddafi at the World Court in a successful legal case which restrained the United States and Britain from taking military action against Libya over spurious allegations that Gaddafi was responsible for the Lockerbie bombing.[2]

source: https://wikispooks.com/wiki/Francis_Boyle

You are right that he is not a virologist or immunologist, but how many of them would know the ins and outs of the legalities of bioweapons development? Probably not many, whereas he literally wrote the main law on the subject. The topic here is the legality of biological processes in the context of bioweapon development and he is absolutely a world expert on the topic.