What are the odds of a 22yo, healthy female getting a septic blood infection through small blisters on hands after handling unwashed chicken eggs&dirt?

Low: First off, if you have a strong immune system, chances are your body will fight off any possible infection from this activity. Second, even if you did become ill, you are not likely to become immediately septic. The first thing that would happen is you would get a skin infection. Hopefully, you would get medical treatment before getting a blood infection.

Depends on the dirt: Where the eggs came from, if she washed her hands afterwards, if she was diabetic, how much alcolhol or drugs she ingests etc. This is a likely story or infection however, so far odds for a wound infection, less for cellulitis, and less still for sepsis.