Clipboard survives a reboot. Is it secure?
You are correct that this information is not secured. This is a very common scenario across most apps on the system when you don't see the keychain dialog to access data. Luckily, there are simple countermeasures you can enact to secure your information. 

Placing a password on your account so that it does not log in upon a reboot will protect your clipboard and all other files on the disk from an idle snooper.

Next, establishing FileVault encryption on the whole drive will stop almost all amateur and some professional attacks as long as your passcode isn't available or guessable. 

Lastly, a screensaver lock and timeout will reduce the window of a sleeping or unattended Mac can be snooped on without knowing your account password. 

There are also apps that clear the clipboard on timed intervals if that truly is your only security worry.