I have some progress thanks to the hints from Steve. I assigned SSL Profile (Client) to the Virtual server, but I'm not sure if I have to do the same with SSL Profile (Client) (we have pools with both and only with the SSL Profile (Client) set up) and now I have this results ffrom curl:
Enter host password for user 'username':
* About to connect() to mysite.com port 443 (0)
* Trying x.x.x.x... connected
* Connected to mysite.com (x.x.x.x) port 443 (0)
* SSLv3, TLS handshake, Client hello (1):
* SSLv3, TLS handshake, Server hello (2):
* SSLv3, TLS handshake, CERT (11):
* SSLv3, TLS handshake, Server finished (14):
* SSLv3, TLS handshake, Client key exchange (16):
* SSLv3, TLS change cipher, Client hello (1):
* SSLv3, TLS handshake, Finished (20):
* SSLv3, TLS change cipher, Client hello (1):
* SSLv3, TLS handshake, Finished (20):
* SSL connection using RC4-MD5
* Server certificate:
* subject: C=US; ST=mystate; L=myplace; O=mycompany; CN=*.mysite.com
* start date: 2012-05-23 00:00:00 GMT
* expire date: 2014-05-23 23:59:59 GMT
* common name: *.mysite.com (matched)
* issuer: C=US; O=Thawte, Inc.; CN=Thawte SSL CA
* SSL certificate verify ok.
* Server auth using Basic with user 'username'
> GET /Servlet HTTP/1.1
> Authorization: Basic dGNsYXNzaWZpZXIxNTp0ZXN0Q2xhc3MzLjU=
> User-Agent: curl/7.21.4 (universal-apple-darwin11.0) libcurl/7.21.4 OpenSSL/0.9.8r zlib/1.2.5
> Host: mysite.com
> Accept: */*
>
* SSL read: error:00000000:lib(0):func(0):reason(0), errno 54
* Closing connection 0
curl: (56) SSL read: error:00000000:lib(0):func(0):reason(0), errno 54