Yes, of course, you can hike to Dudh Kunda on any day of June. But we would still recommend that you don’t until out of desperation or pushed for time.
Monsoon hasn’t much of a good reputation amongst tourists and even locals visiting Everest region. The season brings heavy rainfall with it in and across the country.
Taking it to notice, the trails are often wet and little can you expect to get it dry the next day. Henceforth, it would be best if you plan the trek during March or October instead of June.