If I am shooting a video essay at home, is it better to use a phone or a computer? Which is better for resolution?

The phone is going to be the better option between that or using a webcam or laptop. Much more freedom of movement for both the camera and the subject. You need to have something to hold the camera/phone steady for you, any cheap tripod with a phone holding device will work. Sound will also be better, but an external microphone will work best no matter which one you decide to go with.

Phones don't "scrunch" video. They shoot vertical (portrait mode) if you hold them vertical. They shoot horizontal (landscape mode) if you hold them horizontal, and this is the preferred method as it matches the aspect ratio of an HD TV. The video is recorded at standard TV resolutions, no matter how it looks on the device screen.

These days smart phones actually have a much better camera resolution and microphones than laptops and tablets.  You just need to make sure you film wide (hold the phone sideways), so when the video is transferred to a computer or tv it looks normal when viewed.   To use a laptop or PC you really need to purchase a better usb camera if you are recording videos.  

With a phone you have to take the time to mount it at to get a good camera angle and keep it steady when filming.  

I. would think a table would not only give you adequate resolution but also a wider screen. Phones tend to give you a squeezed up picture