The only way I know of that a birdstrike can cause an engine fire is if it causes blade(s) separation, which then goes through the casing and severs a fuel or hydraulic line (hydraulic fluid will burn but not very vigourously, it goes like sticky brown treacle). They shoot chickens into the intake at max N1 during development testing to make sure this doesn't happen.
[There was a story years ago that a frozen chicken was accidentally used once, I don't know if that's an urban myth though]