I am licensed in Nevada, but this answer should apply in Utah as well.
As a practical matter, I'd like to know how many children were playing when the girl got hurt, how she got hurt and if the other child that accidentally hurt her can be identified. All of these will help determine who is responsible. I'm assuming you and the boy's father are married and living together. That being the case, I would say both parents are responsible equally UNLESS the boy's father was negligent in supervising him. Other adults there may also have acted negligently in supervising the children.
I would also like to know if the little girl had insurance and if the $600. represents all of the medical bills or just the little girls' parents' share.
Now, for the less than practical side......
Is there any relationship between the children (your son, the little girl, and the other children)? If so, why ruin a relationship over such a small sum. I understand that times are hard, but it would be nice if everyone involved were able to work something out - even if it requires payments.
I have a feeling that this is something that can be talked out.
/s Donald Kudler
Was your step-son in some way responsible for the injury to the little girl's arm? If so, depending on the circumstances, you may be liable for the damages she suffered including medical expenses. However, there are many details an attorney would need to discuss with you in order to properly evaluate the potential claims against you. You really should speak with a Utah Personal Injury Attorney in detail about the facts.