theyre going on the assumption that every creature alive today has remnants of their predecessors' dna, even the shit that is unnecessary.
one of the debatable assertions of darwin's original evolution theory is that, as beneficial mutations develop and species evolve, they "lose" the parts of their ancestors that are no longer needed.
but it seems, or it is theorized, rather, that the dna for these unneeded parts still exists today, it's just "inactive".
from what i've heard, they chose a chicken for several reasons. first, its dna has already been decoded. second, the common, non-domestic chicken lacks a lot of the highly-developed adaptations of many birds (i.e. specialized feet, elongated beaks, wings built for long migratory flights, etc). third, chicken eggs are much easier than some eggs to artificially incubate or manipulate. i would assume that their ovum accept artificial injections better than most as well, like how cloning scientists typically stick to sheep and cats because their ovum are hardier.
and anyway it wouldn't be "dinosaurs" so much as it would be showing signs of early bird ancestors, probably not reaching back as far as the pterosaurs, but rather, cretaceous-and-later species, like the enantiornithines. but they are probably hoping for something more like the signs of archeopteryx traits (jurassic period), so they can provide "proof" in the link between birds and reptiles (thereby cementing the theory of evolution further).