I suggest we begin the entry draft June 28 or July 12. SaturdayJuly 5th is a holiday weekend, people go away and do the BBC / party things. usually we do the entry draft 2 - 3 weeks after the actual MLB draft.
The MLB signing deadline for that draft is July 18th this year, so how about a few days after that? The All-Star break the week prior (July 13-15?) would give everyone ample opportunity to do last minute prep and trades. Also, no one would get stuck drafting a player who doesn't sign.