Part of The Secret Garden is set in Yorkshire (in the north of England), so this is probably from that part, since this writing looks very much like it's emulating the Yorkshire accent. I grew up in Yorkshire but I've never heard the idiom about the egg. However the book was written well before my time. Anyway, I can figure out what it means, if that helps. Flirting your tail-feathers sounds like the subject is a prideful person. Therefore "as full of pride as ..." is going to be comparing how prideful he is to a large value rather than a small value (contrast that with the idiom "he's as much use as a chocolate fireguard" which implies that he's no use at all). An egg has no actual meat in it (at least, not what we'd call meat today), but since the yolk and the albumen are highly nutritious, you could think of them as being "meat" at least in the sense of providing as much sustenance and protein as meat. An egg is virtually full up of this "meat". There is a tiny air gap, but for practical purposes we can think of the egg as being 100% full of this "meat", with no space wasted. Therefore how full an egg is of meat is "the maximum amount possible". And that's how prideful this guy is, according to Ben.

I'm curious why you're reading this sort of thing. It certainly isn't well suited for learning English.