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 $\displaystyle f,g$ on $\displaystyle [a,b]$ 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.