The usage "june bug' would be a compound usage, and not a stand-alone usage. Thus, all normal cases where "June" is used on its own are capitalized. In addition, there is no other meaning to "June" (when by itself) other than the name of the month, whereas "march," "may," and "august" all have different meanings when not capitalized.Merriam-Webster (www.m-w.com) reports "june bug" as a non-captitalized usage although it also cites "June bug". Therefore, it would seem to me that "june" fails the "always" capitalized criterion!
That being said, www.scrabble.com fails to recognize it!