Wiktionary [0] seems pretty confident ("Germanic" includes all of the languages below, not just modern German).

[0] From Middle English chiken (also as chike > English chick), from Old English ċicen, ċycen (“chicken”), of uncertain origin. Possibly from Proto-West Germanic kiukīn (“chicken”), or alternatively from Proto-West Germanic kukkīn, equivalent to cock +‎ -en.

Compare North Frisian schückling (“chicken”), Saterland Frisian Sjuuken (“chicken”), Dutch kuiken (“chick, chicken”), German Low German Küken (“chick”), whence German Küken (“chick”), (elevated, obsolete) German Küchlein (“chick”) and Old Norse kjúklingr (“chicken”).

In the same family of languages (proto-germanic), you'll have Swedish for which the word for the meat is "kyckling" which more pronounced like "chickling".

Modern German might have changed along the way, but if you go back far enough, there should be something. I'm not a linguist or a native speaker of a Proto-Germanic language, so I don't know enough to pinpoint this better.

In German, you can also use the word Küken, which is very clearly a cognate of English chicken.

Know this song? https://www.youtube.com/watch?v=I6Yce3AxF9c

Sounds very french-like in origin. Do all in Switzerland use that though? Even the exclusively german-speaking part? In Austria there is a West-East difference, some words spoken by the Vorarlbergers (most western part) we don't too easily understand in the east (it's quite similar to swiss-german in some ways in the western area, but not identical either).