Peel and stick tiles are designed to have maximum contact
with the surface such as plywood. Becker board is porous so will not hold well,
and they will fall off. Priming might help but if paint loosens tiles will come
off. I would glue ¼ in plywood and install tile over that. Just curiosity, why
are you installing becker board under
vinyl tiles?

return the junk to the junk store and start with real, tile / thinset and waterproofing. why put junk into the most important purchase and resale investment of youre life?

To the comment asking why are you installing vinyl tile and not real tile. I wanted to say this, I have had a house full of "real" tile floors in the kitchen, bath, etc., I was in my 30s then and my knees got sore from standing on that hard tile floor cooking dinner after a long day at work and they were really cold in the winter. I have since lived in a house with "luxury" vinyl plank and tile flooring. They are so comfortable, the dog can stand and run on them and they don't scratch. They look beautiful AND unless you really look hard, you can't tell that they are not real wood. The ones in the kitchen are grouted. They look and feel like real stone and my knees don't hurt standing on them--and my glassware and dishes don't break if they happen to hit the floor.