First I want to know.. where is it mentioned that this is true for dimension 1?
To spoil the fun, i suggest us to look at the case when g is the identity (if I understood by "commuting" you meant that the composition of f and g commutes).. I find to find a counterexample.. so I reduce the question to... is there a continuous function f from unit ball to unit ball such that f(x) ≠ x for all x in the unit ball.. Intuition tells me that such a function indeed exists. Consider n=1 and now look at the graph of g(x) = x and consider the box [-0.5,0.5] x [-0.5,0.5], you can easily "draw" a function that lies below the diagonal {(x,x) : x in [-0.5,0.5]} that does not touch the diagonal. So I'm not sure if the case n=1 even holds. Should there be an extra condition here or am I missing something?