Is there a statute or published case law that addresses the rights of a donor over the resulting eggs, sperm, embryo or child? Yes, but only for a sperm donor. (http://codes.ohio.gov/orc/3111.95)
If the statute only refers to sperm donors, is there case law interpreting this statute to provide the same protection in the egg donor context? Yes, the court in SN v. MB (2010), concluded that neither a sperm donor nor an egg donor were the parents of a child. That decision applies to Franklin County, Ohio. There has been no published case law to establish precedent for other counties in Ohio.
There is also an embryo donation statute 3111.97 “Parentage of children resulting from embryo donation” that is written in gender-neutral terms: Both the woman who gives birth to the child resulting from the donated embryo and her spouse (so long as he/she consented) are considered the parents. The embryo donation statute does not cover surrogacy situations.