How do I hold my daughter's pre-school liable for her contracting shigellosis by a care giver?

You are going to have a huge burden here. First of all, they "suspect" that she got it there. That does not mean that she did. Realistically you will have to prove that there is no other way she could have contracted it. You will also have to prove that a worker had it, knew that they had it and had a duty not to come to school because of the risk of infection. An unsymptomatic carrier of the illness will not get you to a judgement in your favor.

Secondly, based on a quick look at Wikipedia, "Shigellosis, also known as bacillary dysentery in its most severe manifestation, is a foodborne illness caused by infection by bacteria of the genus Shigella. The causative organism is frequently found in water polluted with human feces, and is transmitted via the fecal-oral route. The usual mode of transmission is directly person-to-person hand-to-mouth, in the setting of poor hygiene among children.

That definition also causes probelms as it makes it difficult to determine the exact mode of infection. It would have to be one hell of a filthy school to make that a plausable transmission route in my mind.

You might have a cause of action, but it might also be better just to be grateful your child survived what is sure to be one of many ailments she contract at school during her adolescence.