Since $[0,1]$ is a compact space, functions in $C[0,1]$ will all have compact support, so we have $C_c[0,1] = C[0,1]$. The definition of compact support does not require the support to be a proper subset of the domain.
Now, if we want to define test functions $ \mathcal{D}[0,1] = C_c^\infty [0,1]$ then it is not necessarily that the test functions will be zero on the boundary $\{0,1\}$, so how would we define distributional derivative where we want the boundary term to disappear from integration by parts formula, that is (at least for $g\in L^1_{loc})$ $$\int_0^1 g '\phi = -\int_0^1 g\phi'.$$