Take a Cauchy distribution for $Y$: define $Y=Z/X$ with $Z$ a zero-mean Gaussian independent of $X$; then $\eta=YX=Z$ has the desired property; the answer becomes more complicated if $X$ has nonzero mean, as explained here.

Take a Cauchy distribution for $Y$: define $Y=Z/X$ with $Z$ a zero-mean Gaussian independent of $X$; then $\eta=YX=Z$ has the desired property of having a Gaussian distribution and being uncorrelated with $X$.

The answer becomes more complicated if $X$ has nonzero mean, as explained here.