Dissemination of public documents was slow and labor-intensive in 1776. Its too late to know for sure, but its interesting to speculate why a colonial printer hurried, perhaps carried away by the excitement would print copies of the Declaration of Independence and get the date wrong: June, weeks before the document was actually signed.
In the excited aftermath of the Declaration of Independence, colonial printers prepared multiple copies of the document, often for the purpose of public reading.
The John Carter Brown Library has two copies of the version that was printed at Newport by Solomon Southwick not long after the Fourth of July. One copy bears the date July 12. The other, curiously, bears the date June 13, as if it had been printed in advance of Independence Day. Perhaps the normally cautious printer allowed his hands to move too quickly (in patriotic jubilation?) as he set the type for the wrong month.
Explore further: 3Qs: Celebrating America's independence