I totally disagree with needing weights to be the fastest and strongest.

A lot of greats including Ali, RJJ use NO WEIGHTS in their entire workout. I own a book that has the workouts of some of the greats... the author interviewed them personally. Both of the above boxers use absolutely no weights in their workout.

A lot of boxers mention wearing army boots when running. You could do this or use ankle weights.

Sure you won't be able to squat as much if you don't do weights... but you'll be just fine as a boxer... assuming that's what you're after.