Connected sets definition Defn: A set $X$ is connected if there do not exist non-empty, disjoint open sets $U,V$ s.t $U$ $\cup$ $V$ $=X$.
I thought intuitively that this meant that this was like the English dictionary definition of connected- there is no gap in the set.
It is easy to see $(0,1) \cup (2,3)$ is disconnected. Intuitively it seems $[0,1] \cup [2,3]$ is also disconnected- but by the mathematical definition it is connected. I cannot find a $U$ and $V$! 
Also the Wikipedia defn is with closed $U$ and $V$ - how can the two be equivalent?

The open sets of $A=[0,1] \cup [2,3]$ are of the form $O \cap A$ where $O \subset \Bbb R$ is open. In particular, $U=[0,1]=]-1,2[ \cap A$ and $V=[2,3]=]1,4[ \cap A$ are open sets of $A$. Moreover, they satisfy what you require.
For your second question, if $X = U \cup V$ with $U,V$ disjoint non-empty open sets, then $V = X \setminus U$ is closed (and so is $U$), as the complement of an open set. Then $X$ is the disjoint union of two non-empty closed sets, namely $U=X\setminus V$ and $V = X \setminus U$. You can easily prove the converse.
