Pipes burst because of water that has become trapped inside and then frozen. As water freezes, it expands. If too much water is in the pipes, it freezes around valves and joints, where the expanse literally tears the pipes apart. When the water thaws again, the pipes are broken and have serious leaks. They usually need to be replaced to fix the problem.
Frozen pipe problems tend to happen at the lowest points of your plumbing system and near any faucets or fixtures your have, inside or outside of the house. Your boiler does not affect where pipes freeze, just as it cannot affect the temperature of the air, the primary cause of frozen pipes. A broken boiler won't be able to heat water, but it is not the heated water in your house that freezes; it is the water left in pipelines after the boiler has been used.
One of the easiest ways to prevent pipes from bursting is to turn off your water shutoff valves. This is useful if you plan to leave your house for a few days, because it allows you to drain out all water present in the pipes and prevent water from coming in. You can shut valves for each fixture separately or for your entire house in general. Shutting the water to your boiler also prevents frozen pipes -- but again, your boiler is not directly responsible.
Another way you can prevent frozen pipes is to keep the faucets in your home dripping. If water is moving, it has more difficulty freezing. In this case, if your boiler suddenly breaks down, you may have an increased risk of frozen pipes if you are dripping hot water. But you should start by dripping cold water to save on power bills, so even in this case, the boiler should not be a major issue.