The Borg have advanced technology backed by tremendous power. So not only would I say yes, they could borgify Logan (reworking him genetically, of necessary), I could see them assimilating the adamantium into their tech, as well.
The only thing which would really protect Logan would be a Marvel Writer Field, which would assure Logan's mutant abilities exceed the Borg's antimatter-powered, nano-capable technology.
Wolverine has met and battled alternate universe Deathlok incarnations of himself... but writers can do whatever they want with alternate versions of characters without out the burden of adhering to logic or continuity. Also Deathloks are more powerful and technologically advanced then the Borg anyhow.
Wolverine's healing factor should prevent him from being assimilated.