In both cases above the usual cause is your local firewall settings.
Click HERE for a graphical 'Step By Step Guide To Configuring The Windows XP/Vista Firewall'.
A simple way is to use your browser to open a web site that updates regularly (that way it will not be in your local PC's cache) e.g. http://news.bbc.co.uk
Log in and go to the downloads page.
The simple solution is always to delete all settings in the firewall for HexWar.exe. Load the HexWar Game Launcher again and attempt to connect to the Internet again. Most Firewalls will now automatically ask you if you wish to allow HexWar.exe to connect to the Internet where the answer must be 'yes'.
Some firewalls insist on adding the HexWar settings manually. See the section at the end of the page for 'General Firewall Information' setting none Windows XP/Vista built in firewalls. See the Step By Step Windows Firewall Guide to manually set up the built in windows firewall.
The advice from HexWar is that unless you are a techie using a router based firewall is not a good idea. Such a firewall is not dymanic and does not update so can be vunerable to new threats. Using a modern software based firewall is kept up to date by the software suppliers. See the section at the end of the page of 'General Firewall Information' to set up a router firewall.
A common issue is a user having two firewalls turned on. Typically installing a software or router firewall while leaving the built in Windows or Vista based firewall turned on. The built in Windows Firewall should meet the needs of 99% of home users and no other firewall should be required.
Uninstalling and installing HexWar rarely solves anything. If you have a problem downloading the launcher again has the benefit over overwriting any bad files but uninstalling is unlikely to help at all.