Adam7272: Crampons are better, but microspikes might work.

I don't have any experience with the microspikes, but I understand they are not appropriate for descending on steeper slopes in icy conditions. Here's what you should do, if you don't take crampons:

As you encounter steeper slopes while you are climbing, turn around and practice descending. Make sure you are ok in descending mode. If not, then don't climb any higher. Make sure everyone in your party tries this.

None of the Whitney Main Trail route should be too icy, so the micros might work ok. Please be sure to report back what works (or doesn't) for you.
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rpeers209:
By late June, much of the snow will be melted, but there will be some lingering. By then, people are usually using the switchbacks, but encountering a few snowy/icy situations, especially at the cables. People usually just go without any special equipment, but need to be careful in those spots.