I expect June 26 will be a little early for the switchbacks to open up -- they are covered in snow, and until someone breaks a trail through the snow, people use an ice axe and crampons and climb straight up the slope just west of the switchbacks.
Most people do ok with that scenario, but last year, there was a terrible accident where someone lost their footing near the top, and slid and tumbled all the way to the bottom. They were followed by several others! This occurred about 7:30 AM (when it was too cold/icy for the snow to soften and give better traction). It was late afternoon before the several injured were all airlifted down.
With that warning, visit the Reference forum, and especially the Important Whitney Links
post, and read up on the Ice Axe Use topics. Understand that 99% of the people going up the slope do just fine, but there is that added element of risk that you take on.
With all that said, it IS
possible that the switchbacks will be open by late June, and then you would be just fine hiking with microspikes and hiking poles.