Will lobsters caught in June be shedders?

Probably not, the shedding season starts mid-late July according to this site:

By mid-to late July the Gulf of Maine begins to warm up significantly which triggers the seasonal molt.

The end of June is generally not going to be warm enough, however you might get lucky as sometimes its early, and it's worth asking because they are the most amazing tasting lobsters you will ever try. I'd suggest if you're going to be near the coast that you go directly to the source and call your local lobster pounds.

A lobster pound is the place where independent boats bring their lobsters to and sell, from there they go out to restaurants across the world (although some local restaurants will have their own docks and boats). Many pounds are set up as restaurants or cook lobsters and steamers (local clams) to order as take-out. It's the freshest lobster you'll get and the best prices too. Just call them up and ask if they have any shedders and they'll tell you, most will take orders over the phone although you can go to them as well. I hope you find some, there's nothing like them anywhere.