$H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$

You can reduce the problem before you start, as follows: Suppose the path components of $X$ are the spaces $X_i$ with $i\in I$. For each $i$, let $A_i=X_i\cap A$, and show that $$H_p(X,A)\cong\bigoplus_{i\in I}H_p(X_i,A_i).$$ Using this, you are left with proving the statement

If $X$ is a non-empty path connected space and $A\subseteq X$, then $H_0(X,A)=0$ iff $A$ is not-empty.

One way to prove this is to consider the end of the long exact sequence for the pair $(X,A)$, namely $$H_0(A)\to H_0(X)\to H_0(X,A)\to 0$$ By our hypothesis, $H_0(X)\cong\mathbb Z$, and you should know/check that it is generated by the homology class of any point in $X$. If $A$ is empty, then exactness immediately tells you wat $H_0(X,A)$ is non-zero. If $A$ is non-empty, pick a point $a\in A$ and consider the homology class $[a]\in H_0(A)$. The image of $[a]$ under $H_0(A)\to H_0(X)$ is the homology class of a point, which generates the codomain. Exactness now implies that $H_0(X,A)=0$.