street is correct that you would need a special enrollment event to get insurance to start in June. Be careful with the term "lose health insurance". Running out of cobra eligibility when you are using that as your insurance creates a special enrollment period. However, deciding to stop cobra before eligibility runs out does not.
If you have a SEP (special enrollment period), this will open up signing up for insurance +/- 60 days from the event. Typically insurance will actually end when month changes (you have it on the last day of the month, but not the first day of the following month. You likely have to get your insurance forms accepted by the 15th of the month prior to when you wan it to start... and paid before the 1st day it takes effect.
At least I believe this is how it works under current set of rules.