Yes, an unprovoked military incursion into a foreign country is usually a great way to win the population's hearts and minds and ensure its full cooperation.
It's not about winning hearts and minds. It's about containment.
Containment IS about hearts and minds. If you have the trust of the people- if they trust that Ebola is real, that healthcare workers are not out to get them, that they can go to a hospital and receive care- then this outbreak might be containable.
Actually you can, you just need to use enough force and nobody is willing to do that. Ebola could become a threat to the point where the healthy start shooting people trying to escape quarantined areas or simply burning them down.
At the most extreme you could just firebomb or nuke the areas infected. That would be excessive force, but it would solve the problem.
It is difficult, if not impossible, to build trust because the population isn't educated, they don't trust the government from before, and there is no infrastructure for wide-spread dissemination of information.
> You can't win with force alone.
You can.
I think that the part of the problem is that they try to take care of the sick, but they don't have resources, the equipment, the personnel, the education... nothing. The sick can walk out (if they can), anybody can walk in, other people try to "liberate" them, etc. It's mess. Compare their conditions with how ebola-infected people are handled in the west (isolation and containment facilities).
What do to instead: just write off the infected, however cruel that may seem, and transfer them to a closed isolation facility, guarded by military. Heck, I don't think that anything else would be feasible even in any western country if suddenly a few thousand people got infected.
It's not about winning hearts and minds. It's about containment.