The short answer is that CBP has the right to ask to see your electronic devices and you have the right to refuse but if you refuse (and you are not a U.S. green card holder or citizen), CBP likely will deny you admission and send you home.
It's CBP
It's not actually about any form of screening. It's a power trip. Only the dumbest criminal or terrorist would bring an incriminating device across the border and have any need to say "no" to that request.
Which, stopping dumb criminals and terrorists is important and valuable and they do exist, but nobody applied to work for CBP in order to check the phones of dumb criminals. They applied because they have strong opinions about certain things.
An acquaintance used to work airport security and he told me they asked to unlock/power up etc. electronic devices such as phones, laptops or SLR to check if they are real electronic devices. Apparently they were -- maybe are -- afraid of EDs that looked like electronics on the scanners.
Not completely groundless, even if I could think of more than one way to construct a device, e.g. laptop, that boots up and still explodes, but hey.
Perhaps EDs are a concern, but mostly what terrorists actually use phones for is remote detonation. A feature phone is the ideal sort of radio transmitter, modified or not, to send a remote command to detonate explosives. If security is really scrutinizing carry-on luggage, then perhaps the strategy moves to placing a bomb in checked baggage, and carrying only the trigger through the checkpoint.
So if you force someone to turn on their phone or laptop at security, you will hopefully force the decision point; even if the inspector cannot tell the difference, a terrorist is going to get real nervous and jumpy around activating the thing they intend to use as a detonator, in contrast to some businessdude activating his very ordinary mobile device.
For most people, a phone contains:
1. An itemized list of every account they have and their passwords and any 2FA they've used.
2. Multiple forms of payment.
3. A photo gallery that, in many (if not most) cases, will have private content.
4. The contact information of everyone they know, who have not consented to having that information distributed.
I find the entire idea that it's acceptable for any barely-educated border agent having a power trip with no probable cause to demand this under the duress of rejection, repugnant.
Especially in a country where police officers are occasionally allowed to "have sex" with people they apprehend - it seems like there's nothing that actually stops a CBP agent from just stopping any attractive woman they happen to see and demanding their phone so they can find nudes.
Say no? Get deported, no legal recourse.
I had a friend who was asked to unlock her phone, and she did, and then… they did nothing. They watched her do it and moved on to the next question.
Seems like asking someone to do this is just a good test to see the kind of individual they’re dealing with. It’s not practical to thoroughly search phones at scale and plus they know people can just have burner phones anyway. If you’re cagey and combative they know you’re a problem.