Here is where I got lost .. I have a scheme Y over k (an algebraically closed field), in it I have an irreducible closed subscheme X of finite type (do I need finite type?). I also know that X is universally closed (over k) and separated (do I need separated?) .. then why is it that X should be either a point or Y itself?

I don't get it. You are saying exactly that $X$ should be proper over $k$. Embed $X = \mathbb{P}^1(\mathbb{C})$  which is proper over $\mathbb{C}$  in $Y = \mathbb{P}^2(\mathbb{C})$ as a closed subscheme. That's a counterexample? 

