Even though, I'm guessing, Interlaken and Lauterbrunnen will most likely not have snow in mid-June, what about the higher-up Murren?

Possible

possibly not.

Definitely not.
(Even the Schilthorn is reporting 0 cm snow at the moment).

Probably no snow in Mürren itself, but for sure on the surrounding mountains.

Eli is wrong, the Schilthorn mountain (and not the summit, only 2360 m elevation) reported 177 cm and -3 degrees Celsius today. Next week is predicted to be warmer, though, and the snow will melt quite fast.

See http://www.slf.ch/alpen-info/alpeninfo.html for reports on snow/temps in the Swiss mountains. (In German only)

When my husband and I were in Muerren two years ago, it snowed our first morning there, which was June 13. This was snow in the town itself! Then the weather cleared and was beautiful - highs in the high 50's/low 60's in Muerren itself and warmer in Interlaken. I think however, that they had had a rough winter/spring that year for snow and cold. We brought fleece vests and I was glad we had!

We were in Murren the second week of June 2002 for 5 days. Each day was in the mid 70's. We wore short sleeve shirts while hiking. The only snow we saw was on the mountain tops.