It is definitely a CURL issue or firewall blocking curl from accessing external urls. Support people at hosting companies have more interest in getting you off the phone than solving your problem, so you have to make it obvious what the problem is and impossible for them to deny it.
I've created a test page that will demonstrate the issue:
Show that to your hosting company and demand they tell you why it doesn't work and fix it. You can look at the code, it does a simple curl post to get the result from this url:
When you browse that url directly you can see the result, but the same result can't be retrieved through curl. I think that will force them to acknowledge the problem and escalate it to someone on their team to fix it.