Hi everyone,
Two years passed and I still experience the problem.
It made me to basically renounce to Chesscube, but I do like the site, and I give it a try from time to time.
What I can tell is that:
1. It occurs in/with different places/countries/routers/Internet providers/computers.
2. It is not permanent. Sometimes one can play 60 minutes without problem, then there are days in which the engine restarts every 15 seconds or so. What is permanent with it is that it is recurring, and it is unpredictable.
3. It does not seem to be related to an specific Microsoft OS (I have experimented same symptoms on both XP and Win7/64).
4. I use BitDefender (on all systems that I have tried to use Chesscube), I did not disabled the firewall, and I see no reason to disable it

(However, as someone reported bellow it seems to be unrelated to BitDefender)
5. I use Chrome, but I remember that I have tested and it also reproduces on Firefox.
Hope that this info will help finding a remedy!