Will having boils cause the lymph nodes to swell?

Yes: Boils, or skin infections (dermal infections) can as most infections cause local lymph nodes to swell.

Monitoring Centers: Lymphatic system absorbs/carries things too big to go into the arteries/veins like bacteria eaten by white cells and digested food. Lymph nodes "taste" the lymph for anything bad like bacteria/viruses/cancers, etc. When it detects a problem, it alerts the immune system and your body reacts. They enlarge in response to the inflammation they create when they react. They shrink when all is well.