f,g are uniformly continuous on [a,b]
any example f*g which is not uniformly continuous.
and :
example, f,g are bounded but fg is uniformly continuous.
Is this what you meant to write?
Anyway, if you're looking for functions on that are uniformly continuous and such that their product is not, you won't find any. Indeed, on a segment, uniformly continuous is equivalent to continuous (this is called Heine Theorem (at least in France)), and the product of two continuous functions is continuous.