I'd like to know what the pattern of rainfall is in Croatia in June; I read it rains one of every 3 days. Rain patterns vary (tropical 4PM thunderstorms followed by clearing, dreary all day drizzle etc) I'd like to know what to expect. And...will there be mosquitos?

Even though I've been living here for almost 2 years, I can't answer this question. If I could, there's a gorgeous villa down in Cavtat which I would own by now. This spring has been very wet and we all hope that June will be a good month. There's no average, and we are hoping for a warm dry month.
Yes, there are mosquito's and no there are no screens, expect at my home. I believe I am one of the few homes in Dubrovnik with screens on all my windows and both doors. Bring repellent.
Dobrodosli u Dubrovnik

Not a repellant answer though. When it does rain, is it all day or only in bursts?
I hope my Converses don't soak through!

Having lived in Hawaii I can tell you the rain in Dubrovnik is not tropical. There have been times when it has rained all week, all day and a few showers here and there. There is no pattern, it all depends on how the wind is blowing. If we have wind from Africa, Italy rain can happen. However if the north wind, called Bura, is blowing there is no rain.
When you are traveling, you get what you get. There's no distinct pattern around these days. Your chances of having good weather are better in June than April or May.

Dobrodosli u Dubrovnik

Jumping on to humanone's thread, what's the weather like in September? It's great to have a source on the spot in Dubrovnik.

September 07 I was in Dubrovnik for the entire month, it was wonderful. Not too hot, very comfortable and many were still swimming. We left about the 5th of October and everyone was wearing tee shirts and light weight slacks to the airport.
This past September the weather changed into fall about mid month. It was cool enough to wear a sweater over the tee shirt, but it did begin to rain just a bit. However overall it was again wonderful weather and they were some who still ventured into the sea.
From mid September to end of October is the very best weather, not that hot but some rain. Who knows what this year will be, I sure wish I could advise you.

Dobrodosli u Dubrovnik