There are a few meanings behind both sounds. People have already stated some but here's my experiences.
The 'egg song' (or 'bok-bagark' as we call it), is often also used as an alarm call. This is not an issue, until you have a spacky neurotic chook who makes it when there is no issue at all. Some chickens can get mentally trapped in the looping circuit of making that noise all day, every day.
It's not actually too natural nor instinctive for a hen to scream the location of her precious nest to the world. Many birds celebrate laying with singing --- but away from the nest, usually. In this case I believe what we call the egg song is actually a modified instinct caused by human's long history of raiding chook nests. It is after all the natural panic or alarm call they tend to use when predators are in the area or their nest has been raided. I speculate that many hens reacted in that manner to finding humans had taken their eggs, and it's just bred on, like so many incorrect behaviors. Truly smart/instinctive hens do not advertise the location nor presence of their clutches.
I used to reliably know something was wrong and needed fixing when my chooks made the so-called 'egg song' because they didn't make it when laying, but only when predators were in the area. So I'd hear it a total of maybe once a month, or less. Later on, I got newer chooks who were mentally aberrant and liked to repeat this noise over and over and over again, all day every day. They were spacky, stupid, and neurotic.
In no time at all they had the whole flock joining in and repeating this nonstop all day, every day. Times that by 100+ chickens with close neighbours and you see the severity of the issue. It was insane. I managed to train some to shut up on command, but ended up culling the noisiest, and watching what I breed from more carefully. A noisy rooster or hen produces vocal offspring the majority of the time. Also, people repeating their own noises at them can bring about this nonstop noise-making. I have actually had hens who would repeat it until they ran out of oxygen, unable to stop themselves. They were hysterical in the clinical sense of the word and I culled them too. Yeah, I do a lot of culling. It's far quicker and more effective than breeding bad traits out.
About the 'roarrrr' noise they do, as people have noted it means they want something. In this case I think she possibly wants a rooster. My hens don't make that noise unless locked in a cage, which is rarely done to them, and even then they rarely complain for long because I cull for excessive complaints of any kind... Most do not complain at all. The majority of the time, when they make that noise, it's because their boyfriend is absent or I've culled a rooster they were used to having around.
