is it possible to obtain code sample/sample programs showing the
integration of JNI (Java Native Interface) with the RSA sign and
verify functions in Openssl--> using either the EVP api or th RSA /
openssl api?

have you tried asking on the openssl mailing list?
http://www.openssl.org/support/ for the mailing list email addy.

Apologies if you have thought of all of these: 

- are there particular reasons why you haven't used JCE? Open source
efforts such as PureTLS are available: http://www.rtfm.com/puretls/

- does Pete Guttman's crytplib (with its considerable documentation)
provide the functionality you need?
http://www.cs.auckland.ac.nz/~pgut001/cryptlib/

- the only decent book I have heard of specifically on the subject of
SSL is the O'Reilly (surprise, surprise) Network Security with SSL -
no idea if it covers the EVP APIs...