Do signal boosters work if I have only one bar in my home
Yes, so long as the signal booster is compatible with Verizon and you register it with Verizon. A 3rd party signal booster is an independent device and as long as it can pick up cellular signal, will amplify it.
If you are referring to the 4G network extenders sold by Verizon, those work differently than a signal booster and require a wired (preferably cable) internet connection to work. They function as miniature Verizon cell phone towers.
