For the first one let $f(x) = g(x) = x$. Then $f$ and $g$ are uniformly continuous since for $\varepsilon > 0$ you can choose $\delta := \varepsilon$ so that for $| x - x^\prime| < \delta = \varepsilon$ you get $|f(x) - f(x^\prime)| = |x - x^\prime| < \varepsilon$.
On the other hand, $h(x) := f(x)g(x) = x^2$ is not uniformly continuous on $[0, \infty)$ (Proof taken from here):
Let $\varepsilon := 1$. Let $\delta > 0$ be arbitrary but fixed. For $n \in \mathbb N $ let $x_n := \sqrt{ n + 1}$ , $y_n := \sqrt{n}$. Then $|h(x_n) - h(y_n)| = |x_n^2 - y_n^2| = 1$ and $ 0 < x_n - y_n < \frac{(\sqrt{n + 1} - \sqrt{n})(\sqrt{n + 1} + \sqrt{n})}{\sqrt{n + 1} + \sqrt{n}} = \frac{1}{\sqrt{n + 1} + \sqrt{n}} < \frac{1}{\sqrt{n} + \sqrt{n}} < \delta$
for $n$ large enough. Hence you have found a $\varepsilon > 0$ such that for every $\delta > 0$ you have $|x - y| < \delta$ but | f(x) - f(y)| \geq \varepsilon.