You can definitely get some cold temps in June. We were there last year mid-June and woke up to some snow on the ground! It warmed up nicely every day. Leave for the day wearing layers!
FB is a great location in the park, IMO. We were there 8 days using FB as our basecamp from which we drove out and back every day. We had all the food and supplies we needed to make it a week in the park eating primarlily out of RV and picnic lunches. We did eat at a couple places in the park and were NOT happy with many. There was only one place in Canyon, a burger counter that was any good, everything else was overpriced and of poor quality.
The mosquitos at night in FB were pretty rough, bring lots of spray!