If using a third party library is an option, there is jOOU (a spin off library from jOOQ), which offers wrapper types for unsigned integer numbers in Java. That's not exactly the same thing as having primitive type (and thus byte code) support for unsigned types, but perhaps it's still good enough for your use-case.
import static org.joou.Unsigned.*;
// and then...
UByte b = ubyte(1);
UShort s = ushort(1);
UInteger i = uint(1);
ULong l = ulong(1);
All of these types extend java.lang.Number
and can be converted into higher-order primitive types and BigInteger
.
(Disclaimer: I work for the company behind these libraries)