During BCT and AIT what will happen to my health insurance for me and my family?

That entirely depends on your company. Most will continue your private health insurance while you are on active duty orders. Some will require you to continue paying your premiums and others will cover it as part of their 'activated military benefits' - you'll have to check with your company to see what their policy states.

Additionally, once your join the reserves you'll become eligible for Tricare Reserve Select (TRS*) (longer discussion, but in my experience you'll be sorely pressed to find a better Health Care plan ... as long as your doctors take it (most do)). When you are ordered to active duty for more than 30 days, then TRS is converted to the active duty version of the plan (You'll be on Tricare Prime as the active duty servicemember).
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* TRS coverage - https://www.tricare.mil/Plans/HealthPlans/TRS

I went through BCT and AIT a while back, and keeping my family's health insurance intact was a big concern. Luckily, I was on a group health plan through my work with https://www.taylorbenefitsinsurance.com/

They made sure that my coverage continued without any hiccups while I was in training. My wife and kids didn’t have to worry about gaps in their insurance, which took a lot of stress off my plate.