Board Game Arena

I tried to use "Sunday" but it was rejected. Are proper nouns not allowed?

No. In most word spelling games proper nouns are not allowed.
Proper nouns whose spellings are identical to other valid words (verbs, common nouns, or whatever) are allowed: march, may, august, etc.

What may be weird is that some words (month, week day, language names) are considered proper nouns in English, but not in other languages (e.g. Spanish).