The 50-year-old takes over at the Jakemans Community Stadium following the departure of Craig Elliott earlier this month.
Former Mansfield Town boss Cox resigned from his previous post with Kettering Town on Friday evening.
The Poppies were defying their pre-season odds - with Cox leaving them eighth, just a point behind his new team.
Boston's caretaker boss Paul Green will remain in charge for tomorrow's game with Spennymoor Town before Cox assumes full control for Tuesday's game against Farsley Celtic.