You can do post hoc tests with an ANOVA to get at those t-tests you're interested in. The only thing is that in the typical ANOVA setting you're assuming an equal variance. If this really is the case you gain a lot by doing the ANOVA over a whole bunch of t-tests because you get a much better estimate of the shared variance.

But to tell you the truth it sounds like if you're dealing with number of eggs are being laid some sort of poisson regression might be better.