I don't believe it's the case that the Border Patrol has the same authority as police. It depends on the state, and if they have been granted the ability "state peace officer authority" by the state to enforce certain state laws. According to the Border Patrol Inspector's Field Manual (2008), state peace officer authority is:
"Power granted by some states to some Federal officers/agents authorizing them to enforce state law. As separate sovereigns, each state may determine who is authorized to enforce the laws of that state. Usually this means that arrest power is granted to state police, sheriffs, and to various municipal police departments. Some states have enacted legislation designating Federal law enforcement officers as state peace officers with the power to enforce state law. Whether or not a specific Federal officer has this state authority is sometimes difficult to determine."
In any case, they do not have the ability to just order people out of their vehicles as has been stated (well they can order, but the driver doesn't have to comply). Not at suspionless checkpoints at any rate. The reason is because, unlike an encounter with a cop (who is not supposed to pull somebody over unless they have reasonable suspicion of a "crime"), these checkpoints are operated without any suspicion of wrongdoing whatsoever. As such, they cannot order you out of the vehicle. They need to develop reasonable suspicion to further the detention beyond "brief" and they need probable cause in order to arrest or search. Headlights and speed are all "crimes" in the case of cops.
And while the unfortunate Mimms case allows cops to order people out of their vehicles for their "safety" - I would bet that if a cop pulled somebody over without reasonable suspicion (which definitely happens), and the driver has recording equipment to prove that fact, they could refuse an order out of their vehicle. Of course they'd better have a plan as the cop would likely escalate the situation, but I would guess with proof of a lack of reasonable suspicion that any charges against the driver would be dismissed. Maybe this will get pressed to test in the courts in the future.
I personally won't exit the vehicle when I know the cop has pulled me over absent reasonable suspicion.