The first one is correct. The second is not.
You could substitute was for were and it would be OK:
- I would never say that I was superior to a cleaner.
That's just the ordinary past tense, echoing the morphological past of would. If you used will instead of would, was refers to your purported past superiority, while the present am is simply generic:
- I will never say that I was superior to a cleaner.
- I will never say that I am superior to a cleaner.
The reason is that the sogenannte "Subjunctive" is rare in English, and occurs only after certain predicates, of which say is not one, even in a universal temporal negative construction (i.e, never say).
In general, it's best to avoid mentioning the term "Subjunctive" when speaking of English grammar; it simply confuses people because they don't know what it means, either.