If I am on one side of house my phone will connect to it and drop the call I am on if I go to other side of house.
First off, if your neighbor's network extender is set to the signal strength it shipped with, theoretically you need to be within 15 feet of the device to connect to it initially. That is called the network extender's cage. I can tell you that in my experience, my neighbors can connect to mine farther away than 15 feet.
Now once connected, the range of the device is farther than 15 feet by alot, but the exact range is dependent on the structures that the signal has to go through so no one can tell you exactly what the range of your neighbor's extender is unless they come and measure it. And even when you get out of range, what happens is that the network extender is supposed to make a hand off to a cell tower without dropping the call that is if there is a cell tower signal strong enough to hand off to.
All of this indicates to me that you shouldn't be dropping calls walking from one side of your house to the other unless the cell signal is so poor on the other side of your house that the network extender has nothing to hand off to as you exit its range. If so, you would have dropped that call regardless of the network extender's presence since even if there were no network extender, there is not a strong enough cell signal on that side of your house to maintain a call.
What I suspect is happening is that your neighbors can tell you are connected to their network extender because the WAN light flickers when someone is making a call through the network extender and they are purposefully cutting you off by removing the cable from the network extender to their router or cutting power to the network extender. Now THAT will make you drop a call because the network extender is immediately cut off and can't make a hand off to a cell tower.
Bottom line is this - there is no way to password protect a network extender because there is no way for these phones to give a password to the network extender given how they are designed. And remember even if they could put that feature in a smartphone, if existing phones can't do it, then Verizon can't allow it. This is because any cell phone within the cage of the network extender (which in my experience is more than 15 feet) gets overwhelmed by the network extender's strong signal and can't sense a cell tower signal because of it and then preferentially connects to it. That is how you connect to the network extender in the first place. So if they required a password protection, they would be rendering existing phones and basic phones useless within the cage of the network extender because they can't give a password to the network extender.
All you can do is ask Verizon to turn down the signal power on your neighbor's network extender so you won't connect to it. They can do it unless your neighbor's network extender's signal strength is already as low as it goes.
You need to get tech support on the line and they need to put through a network trouble ticket on your behalf for this to occur. Regular customer service reps can't do it. Get to tech support and tell them what is happening and get them to submit a trouble ticket to the network department. Keep calling until you get results. The squeaky wheel gets the oil.