I have the same issue on my Lumia 930 UK, It worked fine when I first got my band about 3-4 weeks ago. I noticed I started getting sync errors on my band last week, (oops message) but the data seemed to be updating so ignored it. I tired the health dashboard website last week for the first time and noticed I could not login, just got stuck in a login loop, again, ignored it with a view to try at a later time. Then recently due to some unrelated issue I had to rebuild my phone , and when I came back to re-install the health app I got the network error and it would not complete the installation.
I tried on my spare android phone and got exactly the same issue, so rules the phone out. The band is OK its seems too. I even installed the windows 10 preview on my phone , but same issues.
Having called MS support they seem to think all the issues above are related and caused by some corruption on my account. They are looking at my account now. As part of the fault finding they asked me to install the desktop app today, that failed half way saying there was no internet connection. However I am on my work laptop and could be blocked by my security software, ill try again on my own PC when I get home.
If anyone has any advice please help.