You may use the eggshell for mtDNA extraction (if you have preserved it as well).
There are several papers on the method
Gregory Schmaltz, Christopher M. Somers, Priya Sharma, James S. Quinn. 2006. Non-destructive sampling of maternal DNA from the external shell of bird eggs. Conservation Genetics
Martín-Gálvez et al 2011. DNA sampling from eggshell swabbing is widely applicable
in wild bird populations as demonstrated in 23 species.Molecular Ecology Resources
Yes definitely you can use the material to extract not only mt but nuclear DNA too, if it is preserved well. In my experience, I had success in extracting mt DNA and amplifying certain genes from 13 years old feather sample that was stored at room temperature.
You can extract mtDNA from unfertilized bird egg , because mitochondria come to the egg from female ( we call it maternal inheritance ) , and male did not contribute with mitochondria ( the mitochondria located at the Mid piece of the sperm )