"If p, then q" = "q unless ~p"? Can anybody please explain to me how "if $p$, then $q$" is logically equivalent to "$q$ unless ~$p$" ?
My problem is with all truth possibilities of $p$ and $q$ except for when both are true.
It seems in those situations they would not have the same logical meaning.
For e.g: "if $p$, then $q$" is true if $p$ is false and $q$ is true. However, shouldn't this possible combination render "$q$ unless ~$p$" false?
Feel free to use truth tables or whatever method that would help in explaining.

Natural languages are ambiguous, so if you want to check a logical equivalence you should "translate" your propositions to the language of some form of logic (propositional logic, predicate logic, ...).  The title of your question is mistaken but then in the description you write it right. "If $P$ then $Q$" is equivalent to "$Q$ unless $\lnot P$" because to make the implication $P \implies Q$ be true when $Q$ is false it must be the case that $P$ is also false (which meand $\lnot P$ is true), otherwise the implication doesn't hold. $Q$ unless $\lnot P$ should be "translated" as $Q \lor \lnot P$ 
