Is there a cholesterol-free substitute for egg noodles?
While I found many references to Chinese egg noodles, I did not find much information on Japanese egg noodles.

Digging into the edit history of your question, it appears that the actual product you would like to substitute for is Plain Tops Noodles.  Their website is parsimonious with information, but does call the product vermicelli, and only indicates that they are made with wheat, and does not mention egg.

Kikkoman, a producer of Asian ingredients for the US market, suggests using suggests substituting Italian style fettuccine or spaghetti for Chinese egg noodles.

Putting this all together, you should be able to substitute dried Italian vermicelli, spaghetti, fettuccine or angel hair in the your recipe with good results.   Check the package to ensure that the ingredients are only semolina (and water), as some Italian pastas are made with egg (this is most likely with the fettuccine shape).  

Any non-egg based semolina pasta will be completely cholesterol free.

See also: Asian pasta reference from Recipe Tips.