Anonytun is a service that probably uses an HTTP CONNECT proxy on the server side to implement most of its functionality. So asking what the equivalent is on Linux is somewhat ambiguous. In any case you will still need a VPN server to connect to.
Tools you could use to make a tunnel are xl2tpd, openssh, apache using HTTP CONNECT over TLS, openvpn, openconnect/anyconnect etc.
On Ubuntu installing network-manager-openconnect-gnome would add functionality to the network manager to connect to anyconnect compatible VPNs with similar usability to that app. Furthermore without any special software you can change your proxy settings to use an https CONNECT based proxy (which is partially a tunnel).
"host spoofing" in the context of Anonytun is a somewhat incorrect way of saying that it provides an override to SSL Server Name Indication (SNI). While this information is not encrypted nor is the destination IP address and so a simple reverse lookup would probably reveal the same name meanwhile "spoofing" it will likely result in unexpected things happening. Since it is an SSL proxy this override is likely for the tunnel itself. In the case of anyconnect for example this information just isn't sent at all.