You won't find "eiz" in any Intel manual, and NASM doesn't recognize it either. x86 has no zero register¹.
This appears to be some GNU-specific syntax meaning "encode SIB byte with no index register". The only case where this would be required by the hardware is when using ESP/RSP as a base register, and every assembler should produce the correct encoding for that if you simply write [ESP].
So using "eiz" on GAS lets you control what is put into the (unused) scale field. One might call that a feature, but it is a meaningless encoding detail similar to which variant of "register to register" opcodes is emitted, something that I don't think any assembler gives you control over.
¹ except maybe on the microarchitectural level, but that isn't visible to the programmer
You can also abuse the displacement math with eiz:
Or some useless prefixes may work: