The red-wrapped eggs symbolize happiness and prosperty, and if an even number of eggs is distributed, that's the way the family is announcing that "It's a boy!" Odd numbers are for girls. I could ponder on this for a moment, but I won't.
Of course, being the hypersensitive infertile that I am, I had to wonder if giving someone good eggs was supposed to help with fertility. Good eggs = good eggs. Or, if I ate them if it would stimulate some kind of Chinese medical reaction that would help me get pregnant.
However, I tend to think that the only thing that these pickled eggs would give me is a bad taste in my mouth and a stomach ache.