After having very slow DSL the past few days at my parent's house I decided to take a crack at it before calling a Verizon tech, as it takes them more than 2 weeks to dispatch to their location. I opened up the box and found that the 3M scotches almost released the wires which they have before, and my parents had to go an entire two weeks without phone and internet.
The orange caps seem to be popping out on their own once or twice a year cutting off the service because the wires are disconnected that way. In this installation it's been very unreliable for them.
Is there any better way to junction these wires or are these scotch connectors the only way to do this?
It's pretty common for Verizon to use those to couple wiring together. If there is too much strain on the cable, it's possible that is the reason for the wires to be disconnecting within the Demarc. Those 3M Connectors crimp onto the jacket of the wire and bridge the copper inside, so unless the connection is being done badly they should hold.
Perhaps asking VZ for a newer NID may help. The newer NIDs have some strain relief built into them under the Telco Side, and also provide a test jack for future service trouble. On a second look, that doesn't even seem to be hooked up correctly. Those terminals in the demarc should be for connecting the wires, and providing lighting protection for the wiring.