Why are break-even and shut down points based on average cost?
When you are finding the break even point, you are looking for a price-point.

The price is essentially the average (and marginal) revenue. Suppose price of good $x$ is $p$.

Revenue = $p * x$.

Average Revenue = Revenue / $x$ = $p$.

So what you are really doing when looking for p is to set Average Revenue = Average Cost. 

At that point, you will also have the fact that Revenue  = Cost. So your logic carries through. You still break even at the point where revenues exactly cover costs.

To see this, note Average Costs = Costs / $x$.

So Average Costs = Average Revenue $\iff$ Costs / $x$ = Revenue / $x$  $\iff$  Costs = Revenue

The interpretation when focusing on average costs is only slightly different. When talking about averages, we're just talking about prices and costs per unit. So you break even at the point where your revenue per unit equals your cost per unit. It is necessarily implied that you also break even where your total revenue equals your total cost.