Military experience is usually a benefit for those who serve for a lot of reasons. Military police would be even better. Yes, there are hiring benefits as well.
The degree will probably be more beneficial in the long run. As stated it will help with promotions. It will also help you become a more educated PO, therefore possibly a better one that you would have been without the degree.
Here in MA most all departments require a 4 year degree to apply. Most dept are also part of the Civil Service program which administers the statewide test for entrance and all promotions in police work. Higher the score, better your chance. There are added points and consideration for vets.
There are rather nice monetary benefits that improve your salary for the rest of your career for each level/degree you obtain also, the Quinn Bill.
1. Military
2. Military police
3. 4 year degree in criminal justice
4. Qualification test if required in your state.
5. Serve as part-time or reserve officer if possible.
6. Become a police officer.
Good luck.