Can you distinguish encryption methods from the output?
Well, with standard protocols (TLS, SSH, IPsec), there really is a blatant unencrypted "I need to login from scratch, please!" message.  That's because when they first come up, they don't share keying material with the peer, and so they can't encrypt any messages.  Instead, they perform some sort of key exchange process using nonencrypted messages, and those messages are easily distinguishable.

Now, I suppose it would be possible to define a protocol where this initial message would be indistinguishable from an encrypted message.  However, the details would appear to be rather finicky (you'd have to worry about message timing and message length), and probably end up being expensive.  In addition, reboots are not actually instantaneous, and when you are rebooting, you obviously can't send out any messages.  I can't think of any scenario where the effort would be worth the cost.