My 2010 (engine replaced a few years ago) has had brake booster issues for a while. Lots of clicking and booster "purr" more often than it used to. One mechanic told me that not replacing the booster will damage the engine. Another said there's no connection. I've been delaying replacing the booster because of the age of the car and hesitation about whether it's worth the cost but if it's hurting the engine I might make it a priority.
No it won't hurt the engine. The brake booster is usually ok even with frequent operation at least until it codes. At that point there is a limited grace period until the brake booster could hurt you through poor braking performance.
Damage to the engine is indirect, say if the brakes don't stop the car and there's something jaggy and hard in front of it that damages the engine.
