Almost anytime is a good time to visit Glacier Park!!
Mid-June will be a bit early for many activities in the higher elevations or backcountry. The Gong to the Sun Road very rarely opens completely before July 4th, so you wont be able to drive all the way to Logan Pass from either side, but you will probably be able to drive parts of the road from either side.
Trails in the high country will be closed, so any hiking will be limited to the lower areas, which may be quite muddy but it also may provide great opportunites for viewing wildflowers. Weather in June can be iffy and can be rainy, but you may also get some very nice weather and it will be very unlikely that you'll run into any oppressive heat. The crowds will also be minimal.
If you are specifically wanting to hike the backcountry or the high country, then mid June is too early, but for almost any other activities June would be a good time to visit.
June can be iffy weather-wise. Sometimes Logan Pass is closed because of snow. If you can wait I would recommend visiting sometime from mid July to the end of September. The advantage of June though is there are fewer people. Hiking is the best way to see the park and there are some amazing hikes there!