I'm not a native speaker either; so, I guess, I'm half forgiven!
In my humble opinion, both could be used here. My conclusion is based on what I heard when I lived in the US.
As an AmE speaker, I agree. Both could/would be used here in the U.S. What you would typically hear/read in the U.S. is "I am a student. I work weekends."OR (A) Hey, why don't you join our rugby team? We play every Saturday." (B) Sorry, I can't. I am working weekends. (No preposition/no article in either statement)