Normally this error means that a connection was established with a server but that connection was closed by the remote server. This could be due to a slow server, a problem with the remote server, a network problem, or (maybe) some kind of security error with data being sent to the remote server but I find that unlikely.
Normally a network error will resolve itself given a bit of time, but it sounds like you’ve already given it a bit of time.
cURL sometimes having issue with SSL and SSL certificates. I think that your Apache and/or PHP was compiled with a recent version of the cURL and cURL SSL libraries plus I don't think that OpenSSL was installed in your web server.
Although I can not be certain However, I believe cURL has historically been flakey with SSL certificates, whereas, Open SSL does not.
Anyways, try installing Open SSL on the server and try again and that should help you get rid of this error.