Plenty of ways to "fix" this. It looks like your in-camera overlays are only 1 pixel wide, so that will make cleaning them up a fairly easy task.
-My first choice would be to use the plugin "reFill" by reVision (company that makes Twixtor). You'd precomp your video and either mask out the overlays or create a solid above it to track matte it out. Then in your main comp, the reFill plugin would be used to fill the missing alpha from the footage. reFill does a pretty great job of filling in cutouts in video. I use it a lot when I've manually stabilized videos with heavy shake when I don't want to crop and scale to fill the frame back in.
-That brings me to the most obvious fix which would be to crop out the edges to get rid of your large rectangle. Then you could try using the brush tool to get rid of the center crosshatch. Try to keep your brush diameter small so it just barely covers the marks.
-Depending on whether the footage is handheld or locked down on tripod, you could take a clean still from your uncompressed video and use the healing brush in Photoshop to create a thin overlay to cover your video track. Problem with that is obviously you've got hands moving in and out of frame under the overlay. You'd have to manually keyframe some brush healing in AE for those parts.
-You could try the precomp method from above and then duplicate the video track below it and use some creative resizing to fill the holes. Perhaps if that leaves a not so great result, you could use an adjustment layer to blur it out just a bit where your overlays are to smooth things out.
-If the footage is handheld, you could precomp and mask, duplicate, and then offset your time so the previous frame acts as the fill. Could also try a combination of the built-in effects like time-offsets and echo. You'd have to do a bit of experimenting.
Another vote for Re:Fill, but you can also try the CC Simple Wire Removal included with AE - four instances of the effect for each line.
Thanks to both of you for your suggestions. I'm experimenting with a few things including CC Wire Removal. I'm not sure how to generate the matte that Re:Fill needs. The documentation just says the tool "uses common settings to help you construct, display and fill holes in your imagery" and doesn't elaborate. How do you do this?
Also, will this really work even when actors' faces are moving through these areas? I'm trying on a sample clip now, but I'm not sure it's working well enough (I'm using CC Wire Removal).
You don't really have to generate a matte for Re:Fill - just as long as your masked video is precomped so that it is missing alpha in your main comp for the plugin to fill. I believe the setting would be to fill alpha and then I usually use something like "blended, nearest neighbor" as the fill option. But those are best left for experimentation, since sometimes "mirror" works as well. I usually add a bit of noise too, which should be one of the plugin options. You just want to match the grain of the existing footage so it doesn't look too smooth and blurry.