While there may well be some things that, according to their chemical analysis, you might be able to use 'off' eggs for, I can can guarantee that you wouldn't want to!
'Bad egg' gas (hydrogen sulphide) is what is inside 'stink bombs'. If you've ever sniffed it, you'll know that it's one of the most repulsive (and nausea inducing) smells that you could ever come across. (Also, despite the fact that joke shops used to be able to openly sell 'stink bombs', it's actually highly toxic).
If your eggs are 'off', you certainly won't want to be cracking them open, irrespective of any possible beneficial uses for their contents!
Chris