From what I can tell, Sun JRE/JDK's are limited to only providing 128-bit strong cryptography without downloading an unlimited strength jurisdiction policy files. My question is, if I install theses files, is there a PRNG with sufficient degrees of freedom to generate 256-bit numbers built into the JRE/JDK?
Java's cryptographic layer is pluggable: you can configure extra providers and even set them as "default".
java.util.SecureRandom, the default implementation (called "SHA1PRNG") that is shipped with Sun/Oracle JVM uses SHA-1 and a 160-bit internal seed obtained from the operating system (see this analysis). For all intents and purpose, its security level is "about 160 bits", which is more than enough.
If you really made enemies among some major deities that have access to sufficient divine computing power to break 128-bit keys (and who also, for some reason, prefer not to strike you with lightning, as is customary), then you will need another PRNG provider. I don't know of any ready-to-use JCE provider that includes a PRNG algorithm that would fit your 256-bit criterion, but such things may exist nonetheless, or could be implemented with relatively little effort.