How things might have been done better.

To me, the main problem of this film was that the main conflict the viewer is subjected to is the constant argument between the 2 main characters. A bickering couple is annoying in any film, but when it's repetitive, meaningless and constant talking-over each other, it only serve to annoy your audience. It might have been better to have internalized the conflict within one or both of the characters.
For example: Instead of Nathan being the driver, and constantly arguing with Samantha, only to go along with her plan just to, basically, shut her up: I would have had Samantha driving the car. This way, Nate is staying with her ONLY because he wants to make sure she doesn't get hurt. The ONLY time that actually happened in the movie is when they went into the house. That would change the conflict from them yelling at each other to his dilemma of 'leaving this to the cops vs. Making sure Samantha is OK'.
Now the conflict becomes something less annoying, and makes the viewer almost sympathetic towards Nathan. If there needed to be a point of argument, then Caleb (the cameraman) could easily make a snotty comment, just to receive a 'Shut up, Caleb'. Also, it would have been interesting to see how they could have handled needed to get gas, or getting food while on the move. Maybe something like having to stop and then find him again using the footage they recorded, etc. There were plenty of better sources of conflict then the two arguing.
Now, as far as keeping the camera on - there rarely is a good reason in these types of films to record everything. And the writers could have put a very good reason into this movie - Have the abductor and his victim switch cars. This way, the Amber alert is now inaccurate, and it gives a reason for the main characters to follow and record - to document that they person they are following is indeed the person in the amber alert. This act would also give a good reason for the police being stalled in their actions to help.
There weren't too many 'lapses' in time, but when there were, they were pretty big jumps. I might have done several jumps, but smaller lengths between jumps. With the end result being something like 'hourly updates' or recording something important. That way, it isn't a case of someone sitting in the back seat of the car, recording his sister crying, or dude eating a sub.
Maybe future filmmakers will think along these lines, and we'll get a 'shaky cam'hand held' movie that doesn't annoy us with their repetitive lines and horrible conflicts, and we'll finally be able to see the true potential in these types of movies. I've seen real potential in this film, Cloverfield, and Monster. I just hope that the breakthrough film comes sooner than later.