Late June will probably present a problem going over Glenn Pass. The snowfall this year has been incredible, with many April storms still dropping snow at 5000 ft. That said, Glenn Pass should have some snow. There are some exposed areas that are not that difficult to go around if there is snow. If you don't know how to use an ice ax, now would be the time to learn. Now I am not saying that the pass will be full of snow, it's just that with the snowfall this year, it could present some problems. That said, this loop is one of the classics of the Sierra's. One could do this trip in 3-4 days and enjoy themselves the whole way. Basically you start at 5000 ft and climb to the top of Glenn Pass at 11,000 something ft. and then it is downhill all the way back to roads end. The Rae Lakes basin is soooo beautiful. There are so many side trips to take. With 7-10 days you should really be able to take in many sights. Dragon Lake is a short 1.5 mile hike on use trails from Rae Lakes, 60 Lakes Basin could be done in a loop for a day hike, and Vidette Meadows offers many oppurtunities along that vicinity. Also you have East Lake and Lake Reflection to try if you don't mind adding some miles to your trip. Oh and Kearsarge Lake under Kearsarge Pass and Kearsarge Pinacles is absolutely stunning too.
There is a ranger station located at Rae Lakes. I would possibly check with them before going up Glenn Pass. I am just erroring on the safe side. Please don't let me deter you, this is an incredible hike. Hope you go.