I can't think of any information being sent that would need to be secure.
It is not necessarily because the information you exchange with nuget.org contains anything secret and thus needs to be secure. By using using SSL you will be certain that it actually is nuget.org you're talking with. Without SSL, somebody might in theory be feeding you bogus packages, and that might be a security problem.
As for the issue you're experiencing with "Could not establish trust relationship for the SSL/TLS secure channel", we've had a similar problem when we started using a new build server:
If you look at the SSL certificate presented by https://nuget.org/, the certification path is: GeoTrust Global CA > RapidSSL CA > *.nuget.org
GeoTrust Global CA was missing as a trusted CA on our new build server, so the problem was easily solved by adding them to the build servers list of trusted root CAs (using the MMC console with the Certificates snap-in).
Update:
On a later service, I've experienced the same SSL issue, and adding GeoTrust as a trusted CA alone didn't solve the problem. In addition, the server was also missing to root CA for https://go.microsoft.com/, which is Baltimore CyberTrust Root (go to https://microsoft.com, and you'll be able to view and download the certificate). Adding this to the servers list of trusted root CAs solved the issue.