What is the difference between non-reflexive or irreflexive? What is the difference between a non-reflexive and irreflexive relation? Is one stronger than the other (i.e. all non-reflexive relations are irreflexive or vice-versa)?

Is this a trick question? [Original question: What's the difference between "reflexible" and "irreflexible"?] None: in both cases, there's no such thing :)
You mean "reflexive" and "irreflexive".
A relation $R \subseteq A \times A $ is reflexive on $A$ if $aRa$ for every $a\in A$. Thus $R$ is not reflexive on $A$ iff for some $a\in A, \text{not } aRa$.
A stronger condition than "not reflexive" is irreflexive. $R$ is irreflexive on $A$ iff for all $a\in A, \text{not } aRa$.
If $A\neq \emptyset$, then "$R \text{ is irreflexive on } A$" implies "$R \text{ is not reflexive on } A$", but the converse is not in general true unless $A$ is a singleton.
