The action $G\curvearrowright\beta G$ is continuous iff $G$ is discrete, so for nondiscrete groups it is not true that $G$ is extremely amenable iff this action has a fixed point.
What one should look at instead, is the biggest compactification on which $G$ acts continuously, this is known as the Samuel compactification, $S(G)$, and can be constructed as the spectrum of the algebra $RUCB(G)$ of right uniformly continuous bounded functions on $G$. This is where the uniform continuity condition comes from. It is then true that $G$ is extremely amenable iff $G\curvearrowright S(G)$ has a fixed point. (Of course on a discrete group all continuous functions are uniformly continuous so $\beta G$ and $S(G)$ are isomorphic)
Note also that if you write explicitly what it means, for a fixed $f\in C(G)$, that the orbit map $G\to C(G)$, $g\mapsto gf$, is $\|\cdot\|_\infty$-continuous, you'll end up with a uniform continuity condition on $f$!
Pestov wrote a whole book on the subject, called Dynamics of Infinite-dimensional Groups: The Ramsey-Dvoretzky-Milman Phenomenon, it is a great introduction to extreme amenability which contains all the details of the above and much more
