Many thanks for everyone's contribution to this thread. I finally have got an SSIS package to execute, thanks to the responses above regarding the 32 bit execution and its options at runtime. I also agree: there is a working 64 bit Oracle client for Windows:
Oracle Data Access Components for Oracle Client 11.1.0.7.10 Beta
http://www.oracle.com/technology/software/tech/windows/odpnet/index1110710beta.html
I did install this on my Windows 7 workstation, a 64-bit host, and have finally connected to the Oracle server, also a 64-bit UNIX server, but as others have noted, after much research and error resolution. The first errors were very vague, complaining about name resolution of the Oracle server. May I suggest also adding a system variable TNS_ADMIN to the [client] operating system? After that, I was at least able to connect to Oracle from the Windows host, of course leading to the package execution issues others on this thread kindly helped to resolve.
Here is the detail on the system variable needed after installation of the Oracle client:
Add an entry to the System Variables of Environment Variables, in Advanced tab of the System Properties in Windows.
VARIABLE = "TNS_ADMIN"
VALUE = "C:\app\mkelly\product\11.1.0\client_1\Network\Admin"
This path has a folder called "Admin" and that subfolder has the critical connection file "TNSNAMES.ORA", with the connection strings to all Oracle environments.
I am pretty sure I needed to reboot the machine before the new variable is recognized by the operating system.
Here is a reference on the use of system variables that help locate the Oracle client: