I use those waterers also for small chicks and haven't had anything like that happen... but I also don't put them directly on the shavings, if I have shavings in there the waterers go on metal pans or bricks. Besides a chick trapped in the back somehow managing to dig under the waterer...
So very sorry about your chick :hugs What sort of water containers do you have / can you post a picture of your set up? I'm having a hard time picturing a chick being able to trap itself under the plastic waterers I'm familiar with.
And Hello :frow and Welcome To BYC!