"I like eating noodles with chicken."
Because the noodles are the main ingredient they are "modified", bot in a literal and grammatical sense, by the appositional words that precede it. Chicken noodle is a dish with a smaller serving of chicken on a bigger serving of noodles. We can imagine, though ridiculous, eating noodle chicken and having noodle either noodle stuffed into a chicken or thrown all over it.. lol
It is correct as it is, because "面条和鸡肉" sounds like two different dishes or ingredients. Here you would like one dish, a combination of noodles and chicken. In the Chinese logic the distinctive part comes at the front, and the main part (that is the basis for many dishes e.g. rice, noodles etc.) follows.
But "noodles with chicken" sounds like two different dishes or ingredients too! It's more common in English to say, "chicken noodles" if you were going to prepare them together as one dish. It is not just in Chinese that the distinctive part should come first, but also in English.