I had to laugh at your post....because the same thing happened to me, TOO!! My first layer sang and sang and sang, mostly bwaaking, but occasionally the egg song...drove me NUTS! Like you, I kept going out there, checking, nothing. After about the third day of this nonsense I finally said "that's it, stick a fork in me, I'm done!" and I quit checking.
It wasn't 4 hours later she did it again, I made myself stay in the house...I'm not going out there....NOT! I went out, peeked in the nest boxes, and there was the most beautiful little light brown egg I ever saw!!
So just about the time you're ready to give up, they'll fool you and lay their first egg.....
Best of luck to you!! Let us know!