Changing the wire to 12 gauge will probably not fix that problem.
While you are at it, you should make sure all the connections and set screws, including for the neutral and ground wires, inside the breaker panel are tight. If you loosen each a quarter turn and then retighten, you will clean and improve the connection a little. (Do not use tremendous force)
Also uinsnap and replace each breaker (while it is switched off) to clean the contact underneath. If a breaker seems loose you may need to move it to another slot to get an improved connection.
Tornado victims: Do not rush to rebuild. Take your time and look for and get a good contractor. Or consider selling the property and moving to a home that is ready to live in.
It not unusual for motors to pull substanicially more than there normal
running current when they first start up.
So your electrical system is not suppling enough current for this.
But it could also be a problem with your electrical suppliers system.
If you start at the boiler and work backwards with a volt meter,
watching the volt level when the boiler kicks in,
you might see where the current limit is.
If it still happens at the service entrance,
then its your electrical companies problem.