Why is overflow interacting with z-index?

The reason the cyan box appears only when overflow-x and overflow-y are visible, and disappears otherwise, is simply because the cyan box is overflowing the cell box. overflow: visible simply means “paint this box even if it is overflowing its containing block” — the cell box is the containing block of the cyan box because its position is relative. Any other values of overflow cause overflowing content to be clipped from view. There is nothing special going on here; in particular, the z-index is completely irrelevant and there is no such interaction as the question title alludes to (and there really is no reason to set it to such a huge number unless you’re worried about scripts injecting other elements into the cell).

The only way to allow the cyan box to appear while the cell has a non-visible overflow is to remove position: relative from the cell and apply that declaration to the parent of the cell (in your example, it’s the body). Like this:

body {
  position: relative;
}

.boxy {
  position: absolute;
  z-index: 9999;
  top: 70px;
  width: 50px;
  height: 50px;
  background: #0FF;
}

.cell {
  border: 2px solid #F00;
  overflow: auto;
}

<div class="cell">
  Here is some text to keep things interesting
  <div class="boxy"></div>
</div>