Start checking the 10 day forecasts every day before you leave, there are many of them and check for Ronsevalles, Pamplona and little far down the route. This time of the year there is no clear pattern and it changes quickly. I always tell friends if you watch the North American weather pattern add a few day for it ot pass across the Atlantic that's what you might encounter in Navarre. Good example would be this latest cold spell in the NE, it will be cold in NW Spain late this week.
June is normal nice and your layers might not require something that heavy. Rain is always possible but I layer with a t-shirt, light weight long sleeve sweat shirt early in the morning and use a poncho in the rain.
Others will give all kinds of advice. So pick your own poison.