Embryonic chicken cells are commonly used in vaccine production. The viruses are grown in chicken eggs, or in embryonic cells taken from those eggs, and then inactivated or attenuated to produce the vaccine. But why can human viruses—some of which are not known to infect adult chickens—infect immature embryonic chicken cells?
For instance, according to this WHO fact sheet (https://www.who.int/news-room/fact-sheets/detail/measles), "Measles is a human disease and is not known to occur in animals." Yet the the measles virus prepared for the MMR vaccine is grown in chicken embryonic fibroblast cells.
N.B.: According to http://www.ogrod.uw.edu.pl/__data/assets/pdf_file/0010/2161/13a.pdf, while measles may have evolved from domestic animals, it is thought to have its origin in cattle rather than chickens: "Measles, for example, is closely related to two other morbilliviruses-canine distemper and rinderpest (a disease of cattle)" 
One guess is that, while a virus would normally require interaction with a species-specific surface protein to initiate membrane fusion (see https://www.frontiersin.org/articles/10.3389/fmicb.2011.00247/full ), and thus viral (or viral RNA) entry into the cell, the naive nature of undifferentiated chicken egg cells enables such fusion to occur without such species-specific proteins.
Another guess (in the opposite direction) is that the species-specificity of viral infection depends on species-specific cell recognition factors, and because the cells in chicken eggs are undifferentiated, or minimally differentiated, they possess a broad range of recognition factors (where the non-chicken-specific ones are lost as the cells differentiate).
It was surprisingly difficult to find information online about this. According to the the following (https://www.jstor.org/stable/30105172?seq=1), chicken embryo cells do become more resistant to viruses as the embryo ages; but this doesn't explain the lack of species-specificity:
"Cells dispersed from young embryos were permissive to viral growth, while those from older embryos were restrictive in an age-related pattern similar to that observed in ovo. The mechanism of natural antiviral cellular resistance did not involve viral attachment or release from cells from older embryos, but apparently depended on intracellular events during viral replication. These observations suggest that increasing natural resistance is based partially on intrinsic cellular changes. Sensitivity of a given virus and responsiveness of cells to the action of interferon may be more important to antiviral resistance than the amount of interferon that the virus induces." 
 Dobson, Andrew P., and E. Robin Carper. "Infectious diseases and human population history." Bioscience 46.2 (1996): 115-126.
 Hashiguchi, Takao, Katsumi Maenaka, and Yusuke Yanagi. "Measles virus hemagglutinin: structural insights into cell entry and measles vaccine." Frontiers in microbiology 2 (2011): 247.
 Morahan, Page S., and Sidney E. Grossberg. "Age-related cellular resistance of the chicken embryo to viral infections. I. Interferon and natural resistance to myxoviruses and vesicular stomatitis virus." The Journal of Infectious Diseases (1970): 615-623.