The deadline for voter registration and requesting an absentee ballot is technically 5:00 p.m. the day before the primary. Of course, if you're overseas, even if you can request a ballot that late, there is not time left for you to return your ballot so that it will be received by the day of the primary. So, the practical deadline for military and overseas voters depends upon the amount of time it takes for mail to get from where you are to Minnesota.
You can submit your absentee ballot application and blank ballots can be sent to you by mail, fax or as a .pdf attached to an email. However, the voted ballot must be returned to your local election official by mail or an express delivery service. So the deadline for voter registration and applying for an absentee ballot depends upon the time it takes to return the voted ballot.