No, child care expenses cannot be claimed as a business expense as it is not an ordinary, necessary and reasonable expense.
You have to claim these expenses in the child care credit section on your personal tax return.
**Mark the post that answers your question by clicking on "Mark as Best Answer"
Childcare is clearly a necessity for those who have children, but it is not an "ordinary and necessary expense of carrying on a trade or business".
(The IRS acknowledges the widespread need for childcare and the Child and Dependent Care Tax Credit addressed that need as lawmakers saw fit).
To meet that requirement, the majority of those engaged in a specific business would also require and pay for childcare. Clearly, that would not be the case since everyone doesn't have children.
To be deductible, a business expense must be both ordinary and necessary. An ordinary expense is one that is common and accepted in your trade or business. A necessary expense is one that is helpful and appropriate for your trade or business. An expense does not have to be indispensable to be considered necessary.
See The Meaning of Ordinary and Necessary Expenses for an exhaustive discussion of this topic.