The statement that you are asking about has two parts:
If $X$ and $Y$ are independent, then $X$ and $Y$ are uncorrelated.
If $X$ and $Y$ are uncorrelated, then $X$ and $Y$ are independent.
Statement 1 is always true and imposes no additional constraints on $X$ and $Y$ other than what already has been assumed, viz. that they are independent random variables. Statement 2 does not hold in general, but it does hold if we constrain $X$ and $Y$ to be jointly Gaussian random variables. That is,
2'. If jointly Gaussian random variables $X$ and $Y$ are uncorrelated, then $X$ and $Y$ are independent.
is a true statement, and so
Jointly Gaussian random variables $X$ and $Y$ are uncorrelated if and only if
they are independent
is a true statement but
"Random variables $X$ and $Y$ are uncorrelated if and only if they are independent"
does not hold in general. Nor is
"Gaussian random variables $X$ and $Y$ are uncorrelated if and only if they are independent"
a true statement. (Note that in contrast to 2'. the word jointly is missing from the statement). For example, suppose that $X\sim N(0,1)$ and $Z$, independent of $X$ is a Bernoulli random variable with parameter $\frac 12$. Set $Y = (-1)^ZX = \pm X$ and note that $Y \sim N(0,1)$, just like $X$. But,
$$E[XY] = E[(-1)^Z X^2] = E[(-1)^Z]E[X^2] = 0 = E[X]E[Y]$$
showing that $X$ and $Y$ are (marginally) Gaussian random variables that
are uncorrelated. That they are not independent is easily see because conditioned on the event that $X = x_0$, $Y$ takes on values $x_0$ and $-x_0$ and is thus a discrete random variable instead of continuing to enjoy the standard Gaussian density as it would have if only $X$ and $Y$ were independent random variables. Note that $X$ and $Y$ do not have a jointly Gaussian density.
Finally, if $X$ and $Y$ are Bernoulli random variables or more generally, discrete random variables that take on only two different values, then the statement
Bernoulli random variables (more generally, dichotomous random variables) $X$ and $Y$ are uncorrelated if and only if they are independent
is a true statement. See this question and its answers for some details.