Embedded JWK keys are public key objects

They are public keys, and json object (not encoded as strings).

https://tools.ietf.org/html/draft-ietf-jose-json-web-signature-36#section-4.1.3
says:

 The "jwk" (JSON Web Key) Header Parameter is the _public key_ ...

(emphasis mine)

It wouldn't make sense to attach the private key in the header of
a JWS.

Examples of JWK public keys can be found at:
https://tools.ietf.org/html/draft-ietf-jose-json-web-key-37#appendix-A.1
Added API for signing JWS tokens
Support for x5t#S256 (SHA256 fingerprint)