The tiles appear to be linoleum - the grid pattern on the back is a jute (natural fiber) mesh. I'm not as certain about the underlay, or even what the composition of Masonite could be.
I've taken out asbestos tiles in a house - the issue is minimizing breakage and dust and keeping the nasty bits as encapsulated or undisturbed as possible. In any case, regardless of your particular situation, I'd go for protective gear just to be safe (respirator, hooded disposable protective coveralls, safety glasses and gloves).