Mid June should be pretty nice. Granted, there can be a freak storm passing through, but that is generally an exception. Weather forecasts are pretty good these days, so you will know what to expect about a week before your hike.
Visit the Whitney weather link
(above on the left on a computer view) There is a weather link on that page that shows temperatures at the Cottonwood Lakes remote weather station. On a computer screen, you can change the temperature graph to show you the temperatures for June of last year. It shows lows around 35F, highs in the upper 50s, extremes are 20 and 70.
Our winter has started out extremely dry, but the past week got us half way to average precipitation for this date. If it stays below average, you won't have much snow to deal with at Cottonwood Lakes.
Now the biggest issue is the altitude. Horseshoe Meadows is at 10,000 elevation, Cottonwood Lakes is about 11,000. Everyone in your party needs to acclimate -- spend some time at a moderate elevation for (at least) a day or two before heading too high. Most people can handle going immediately to 10k elevation, but some are adversely affected. Since you are coming from so far, I hope you plan on staying and visiting other areas -- Yosemite is 3 hours drive from Lone Pine, Sequoia/ Kings Canyon are a long drive around to the west side. If you visit those areas (or even Mammoth or Tahoe) before heading to Cottonwood Lakes, then you will probably be ok.
I hope you have a great trip!