Someone who 'burrows' themselves from real life situations, someone who is afraid of either revealing/facing/accepting the truth, by digging, or 'burrowing' themselves into holes to hide secrets, normally such things like homosexuality. The word 'Bugsy' coming from a common Rabbits name, which are known for 'burrowing'/digging holes.
'Dude, have you heard about Toby? He's gay?'
'Nah he told me it wasn't true,'
'Trust me, I've seen pics, he's definatily being a bit Bugsy.'