One lesser-known tip for iPhones with FaceID or TouchID: press and hold the lock button and one of the volume buttons until prompted for power options/medical ID/emergency call. You'll then have to enter a passcode in order to use those auth methods again. Having to reveal a passcode can sometimes be considered a higher bar than biometric auth. You can do this even when in it's in your pocket without looking, quite quickly, and there is haptic feedback.
Pretty much every country you travel to can detain you at the border simply for being suspicious. You generally have very few rights as a non-citizen and have to decide if you will really risk being detained vs. giving up your phone passcode.
On android you can hold down the power button and then tap lockdown mode. Then it will require your pin/password to unlock again. Ideally have a password as your authentication method if you're concerned about privacy.
On newer Android, holding down power apparently just launches Gemini's AI assistant. You can change it back in settings, or just hold down power + volume up to get to the old menu that has the Lockdown button.
(Not to be confused with power + volume down which takes a screenshot).
Border patrol agents have the authority to compell you to give them your password or passcode. If you refuse, they can detain you and seize your device (same for FaceID/TouchID).
That's assuming border patrol operates within the law and constitution.
This is not really a checkmate. If the border officials are suspicious, disabling biometrics is not going stop them. What are they going to say? "Ohh! Drat! You foiled us! We can't compel you to enter your passcode! Go on, have a nice day!"
I wish there was a way to activate it with Siri though, in case I were accosted in a manner such that I could talk to my phone but my hands were, say, handcuffed.