Can my Jini client use native code that it downloaded from the server?
No, why would it need to? Java provides no mechanism for downloading and installing native code automatically. This would be a major security hole, and counter to the Java philosophy of platform independence.
Clients may use native code on the client side; for example, the java.awt implementation or the javax.comm implementation both consist of mostly native code. Clients may also use server methods that invoke native code on the server side. But there is no way for the server to know what platform the client is running on, and no way for the server to provide platform-specific code for downloading.