When I read that Firefox and Chrome will only allow encrypted HTTP/2 connections, I was shocked and disappointed. The standard itself does not require encryption, so this is an intentional limitation that those browsers chose to add.
Encryption by itself can only be trusted if you can be sure who you're talking to. Otherwise, you could have an encrypted connection to an adversary, who then makes an encrypted connection to the web server you wanted to reach. Because of this, web servers need certificates from recognized certification authorities. Without a trusted certificate, web browsers show scary warnings, as if something is horribly insecure. Firefox also makes getting around that warning annoying. In reality, such an encrypted connection is no worse than an unencrypted connection. It could be better, but you have no proof of that.
Because of these warnings, if you want to set up an HTTP/2 server, you effectively require a certificate from a recognized certification authority. This is an unprecedented limitation! The need to register somewhere to run a web server is reminiscent of what a totalitarian state might do. You generally even have to pay money, as if HTTP/2 is shareware with nag screens if you don't register.
Then I learned that StartSSL is offering free dynamic DNS and certificates. This means you can get a subdomain and associated recognized certificate for free. This gives you a way to use HTTP/2 for free, and you can probably also avoid supplying accurate personal information. It definitely helps, but having just one company in the whole world offering this isn't good.
What finally changed my opinion was thinking about the big picture. We know that there is extensive eavesdropping going on, by entities which have absolutely no respect for privacy. You can choose to use encryption wherever possible by installing HTTPS Everywhere, but many sites still don't support encryption or don't have it properly configured. HTTP/2 is being used as a tool to induce sites to start supporting encryption. What is being done with HTTP/2 may seem wrong, but it is being used to fight against something that is far more wrong. So, I think requiring encryption with HTTP/2 is justified.
Perhaps the best thing here is how those implementing new technology are leveraging that power, and not giving it away.