Assuming you were able to modify the OS/firmware/device for server/client to send and listen on ports higher than 65535, could it be possible to plant a backdoor and have it listen on, say, port 70000?

I guess the real question is this:

If you rebuilt the TCP/IP stack locally on the machine, would the overall concept not work due to how the RFC 793 - Transmission Control Protocol Standard works as mentioned below in some of the answers? Making it impossible to access a service running on a port higher then 65535.

There has been so much talk about hardware and devices having backdoors created that only government have access to for monitoring, and I was just curious if this was possibly one of the ways they were doing it and avoiding detection and being found?

No, the port number field in a TCP header is technically limited to 2 bytes. (giving you 2^16=65536 possible ports)

If you alter the protocol by reserving more bits for higher ports, you're violating the specification for TCP segments and wouldn't be understood by a client. In other words, you're not speaking TCP anymore and the term "port" as in "TCP source/destination port" wouldn't apply. The same limitation exists for UDP ports.

That said, a backdoor could instead communicate over a different protocol than TCP or UDP to obscure its communication. For example, icmpsh is a reverse shell that uses ICMP only. Ultimately, you can also implement your own custom transport-layer protocol using raw sockets that can have its own notion of ports with a greater range than 0-65535.
