No, individuals with XY chromosomes typically do not produce eggs. This is because the Y chromosome contains the SRY gene, which initiates the development of testes in a fetus. Testes produce sperm, not eggs. Eggs are produced by ovaries, which develop in individuals with XX chromosomes.
However, there are exceptions. Some individuals with XY chromosomes have conditions like Swyer syndrome (XY gonadal dysgenesis) where they develop with female external genitalia but have non-functional streak gonads instead of ovaries. https://medlineplus.gov/genetics/condition/swyer-syndrome/ These individuals cannot naturally produce eggs. https://www.ncbi.nlm.nih.gov/gtr/conditions/CN043561/ While they cannot produce their own eggs, they can conceive through egg donation and assisted reproductive technologies. https://my.clevelandclinic.org/health/diseases/swyer-syndrome https://rarediseases.org/rare-diseases/swyer-syndrome/
Another relevant point is that while the majority of females have XX chromosomes and males have XY, there are genetic variations. Some individuals with XY chromosomes can develop female characteristics but still lack the ability to produce eggs naturally. https://novonordiskfonden.dk/en/news/more-women-than-expected-are-genetically-men/ https://www.iflscience.com/the-woman-who-gave-birth-despite-most-of-her-cells-having-xy-chromosomes-61747
Furthermore, research in mice has shown that it's possible to produce offspring from two male parents, but this involves complex laboratory techniques that are not naturally occurring and don't change the fundamental fact that XY individuals don't naturally produce eggs. https://www.nature.com/articles/d41591-023-00031-9
In summary, while assisted reproductive technologies can enable pregnancy in certain individuals with XY chromosomes, the natural production of eggs is not possible with an XY karyotype.
