Yes, I think it is possible, not necessarily a good idea. The manufacturer could bundle the BOINC software with every phone like other software they include (bloatware), but there are numerous problems with getting BOINC to run effectively on a phone. Battery issues, overheating, not being able to use all cores among others. Those who wish to use their phones to run BOINC do so successfully, they opt in to this knowing what they are getting into. It is not as easy as just putting the software on and then just running it without some thought being put into how the phone will be used. Perhaps one of those running BOINC on a phone would like to chime in.

Cheers

It's usually only advisable to run BOINC on a phone or other mobile device when it's plugged in and has reached a certain charge percentage so that you actually make progress on the charging since the amount of power a USB port or charger can provide is variable.

Regarding using one account on other people's devices, you can give out something called a "weak account key" that will allow you to attach a BOINC project to your user account, but not allow access to the account on the web to modify preferences, etc. While there are apps sponsored by some of the phone companies (Samsung and HTC) I don't know if you could install them with a pre-existing account, as that would violate the terms of the BOINC software agreement. You can only install and run BOINC on devices for which you have permission from the owner.