Prove Context Free languages not closed under difference?
Yes I believe it is possible to prove such a non-closure property without pumping a specific example language.

As noted in the comments, complement $\Sigma^* - L$ is a special form of difference. It turns out that complementation is an extremely powerful tool used with context-free languages. One can show that computation histories of Turing machines can be specied as complements of context-free languages. This seems to originated from J. Hartmanis, "Context-free languages and Turing machine computations", Proceedings
of Symposia in Applied Mathematics, Vol. 19, AMS, 1967, 42-51.

A computation history (for a fixed TM $M$) is a string of the form $x\$C_0\$C_1\$\dots\$C_n$ where $x$ is an input string, $C_0$ is the initial configuration of $M$ on $x$, meaning a string of the form $q_0x$ where $q_0$ is the initial state, $C_n$ is a final configuration (with a final state) and each pair of consecutive $C_{i-1}\$C_i$ represents a single step of TM $M$.

The language of those strings is the complement of a context-free language. 

What is CFL would be closed under complement? Then the context-free languages would be equal to the languages accepted by Turing Machines: the languages of valid computations would be context-free, and from this we can find their prefixes or the accepted inputs of the TM. I  do think that is a contradiction without the pumping lemma. We know there are more TM accepted languages than CF languages by complexity or decidability arguments.